The unramified principal series of p-adic groups: the Bessel function by Mario A. DeFranco A.B., Princeton University (2009) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of INS?'P E WUE MAA -TTs ~F O TECHNOILOGY Doctor of Philosophy JUN I 2 01 at the BARIES MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2014 @ Massachusetts Institute of Technology 2014. All rights reserved. A uthor ................. Signature redacted Department of Mathematics May 2, 2014 redacted Signature .. ...... C ertified b y .......... ...... .. ....... ....... (}.. . ... ... ... . ... ... ..e Benjamin Brubaker Associate Professor Thesis Supervisor Accepted by Signature redacted Alexei Borodin Co-Chairman, Department Committee on Graduate Students The unramified principal series of p-adic groups: the Bessel function by Mario A. DeFranco Submitted to the Department of Mathematics on May 2, 2014, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Let G be a connected reductive group with a split maximal torus defined over a nonarchimedean local field. I evaluate a matrix coefficient of the unramified principal series of G known as the "Bessel function" at torus elements of dominant coweight. I show that the Bessel function shares many properties with the Macdonald spherical function of G, in particular the properties described in Casselman's 1980 evaluation of that function. The analogy I demonstrate between the Bessel and Macdonald spherical functions extends to an analogy between the spherical Whittaker function, evaluated by Casselman and Shalika in 1980, and a previously unstudied matrix coefficient. Thesis Supervisor: Benjamin Brubaker Title: Associate Professor 3 4 Acknowledgments I would like to thank my advisor, Benjamin Brubaker, who introduced me to this field of study and supplied invaluable guidance and support throughout my time at MIT. I would like to acknowledge Professor Ju-Lee Kim for a helpful conversation about Iwahori subgroups and Professor Arun Ram for a helpful conversation about unipotent subgroups. I would like to thank everyone at MIT for providing me plenty of time and space to write this thesis. Finally I would like thank my parents, Toni and Mario, and the rest of my family for giving me so much encouragement in too many ways that I can count. 5 6 Contents 1 Introduction 2 The Subgroup Iht and Vectors 3 Defining Integrals 27 4 Computing W(p(a) - 0U,;.) via a Quotient Space 31 5 Intertwining Operators Applied to 37 6 Evaluation of the Bessel Function 51 7 Computing WH(X, w) 55 8 9 # 21 7.1 Statement of Theorem ....... .......................... 7.2 Explicit Bruhat Decomposition ...... 7.3 Paths in a Root System ....... 7.4 Nesting Coordinates ....... 7.5 Final Integration 7.6 The Support Conditions Are Independent of z . . . . . . . . . . . . . 74 ) 95 ..................... ......................... ........................... 7 57 65 68 . 71 ............................ The Matrix Coefficient S(p(g) - # 55 8 Chapter 1 Introduction Let G be a connected reductive group over a non-archimedean local field F with split maximal torus T. Assume that G is defined over o, the ring of integers of F. Let B = TU be a Borel subgroup for G. The simplest class of representations of this group are the unramified principal series - those arising by parabolic induction of a character x of T(F)/T(o). We denote the resulting representation by Ind (X). In this thesis, we examine two matrix coefficients of the representation IndB(x); one is known as the "Bessel function" while the other has been previously unstudied. Recall that a matrix coefficient associated to a representation (p, V) is a function on the group G, built from a linear functional L : V - C and a vector v E V as follows: g '- L(p(g) -v). (1.1) The linear functional is often assumed smooth when working in the category of smooth representations of G(F). We do not assume that here. Matrix coefficients are key objects in the study of p-adic harmonic analysis and automorphic forms and representations. An important step in gaining a deeper un- 9 derstanding of matrix coefficients is to determine their values at specific elements of G. That is, we seek explicit formulas or closed expressions that are more illuminating than the initial definition (1.1). Two prominent matrix coefficients are the "Macdonald spherical function"and the "spherical Whittaker function"; the terms and constructions are explained below. The Macdonald spherical function (which we abbreviate to "spherical function") was first investigated by Macdonald [Mac] and later Casselman [Cas]. The spherical Whittaker function was studied by Casselman and Shalika [CS], who proved a conjecture of Langlands about the precise formula at torus elements. These authors succeeded in establishing explicit formulas for their matrix coefficients in terms of certain polynomials significant in combinatorics and representation theory. This thesis continues the pursuit of such explicit formulas. The main result is the evaluation of the Bessel function at torus elements of dominant coweight in terms of representation-theoretic polynomials. To our knowledge, it had not been recognized in the literature that this function should even have an explicit evaluation. This thesis also makes apparent an intimate connection, or analogy, between the Bessel function and the spherical function; this connection had also been unknown. Specifically, the Bessel function shares important properties with the spherical function that were described in [Cas]. We subsequently consider another matrix coefficient, previously unstudied, and show that it is connected to the spherical Whittaker function in such a way that extends the analogy we have demonstrated between the Bessel function and spherical function. The matrix coefficients mentioned above are constructed by mixing and matching certain linear functionals and vectors of Ind (X) that are invariant under the action of certain subgroups of G. These subgroups are K = G(o), a maximal compact subgroup of G, and a unipotent radical U of a Borel subgroup of G. 10 First, there is the K-invariant functional S, known as the "spherical" functional in comparison with the archimedean setting, where K = SO,(R) is a maximal compact subgroup of G = SL,(R). It satisfies S(p(k) -v) = S(v) for all k E K and v E IndB(x). Second, there is the (U, 0)-invariant functional W = W(U, V)), where 0 is a nondegenerate character of U. This functional is called the "Whittaker" functional, again in comparison with archimedean case where matrix coefficients made from this functional are associated to a differential equation due to Whittaker. It satisfies W(p(u) - v) = 0(u)W(v) for all u E U and v E IndB(x). For an arbitrary irreducible smooth representation, it is not clear that non-zero functionals with these properties exist. However, it is true for unramified principle series. We note that (K, 1) and (U, 4) are examples of generalized Gelfand pairs: given a a subgroup H of G and a character 0 of H, recall that (H, 4) is a generalized Gelfand pair for G if the induced representation IndG(0) is multiplicity free. This ensures that any G-module homomorphism from V to IndG(,0) is unique up to scalar and this fact may be exploited in applications. The invariant vectors in Indi (x) used in the construction of the matrix coefficients are the K-fixed vector #K, known as the "spherical vector", and the (U, 4)-invariant vector #e, known as the "Whittaker vector". They satisfy p(k) -$K =OK for all kE K=G(o) 11 and p(u) - bU,, = P(u)#U,, for all u E U. For an arbitrary smooth representation of G(F), it is of course possible that such invariant vectors do not exist in the representation. However, they do exist for unramified principle series. Thus we arrive at four possible matrix coefficients for Ind (X) formed from these choices of functionals and vectors. The spherical function is the matrix coefficient constructed from S and #K, and the Whittaker spherical function from W and #K- As stated above, this thesis considers the Bessel function, constructed from W and #gO; it gets its name function in the archimedean case which satisfies a differential equation due to Bessel. We also evaluate the matrix coefficient from S and OUO which has been unstudied. In the remainder of this introduction, we discuss known results and methods of proof. We summarize some appearances of the Bessel function in the literature. The asymptotics of Bessel function on p-adic groups are studied in the thesis of Averbuch [Ave]. Cogdell, Kim, Piatetski-Shapiro, and Shahidi use the Bessel function to obtain stability of local -factors in [CKP-SS] and [CP-SS. They also use "approximate Whittaker vectors" as a way to define the Bessel function. In addition, Gelbart and Piatetski-Shapiro use the Bessel function to study e-factors. Baruch [Bar] and Lapid and Mao [LM] study general properties of the Bessel function integrals. The matrix coefficient construction using a vector in Ind)BG(X) appears in Baruch and Mao [BM] and Soudry [Sou] but they do not name the vector they use as the "Whittaker vector". The authors of [BM] restrict their attention to rank 1 groups and their double covers; we note that the the limiting process used there to define the integral of the matrix coefficient is generalized in this thesis to arbitrary G. These authors 12 do not seek explicit formulas for these functions on p-adic groups. Finally, in [P-S, Piatetski-Shapiro employs a vector similar to 4 in the finite field setting, but there it is called the "Bessel vector". I first encountered the notion of the Whittaker vector while examining [GKMMMO] which deals with constructions of archimedean Whittaker functions. After computing the corresponding matrix coefficient in the p-adic setting for certain groups, I conjectured the explicit formula for arbitrary G. Upon learning the similarity with the formula for the spherical function, I examined Casselman's paper [Cas] to see if there was a connection. This led me to seek analogues of the objects in [Cas]. We briefly compare [Cas] and this thesis below. First we describe the setup of the main result, Theorem 1.0.2. The root datum (X, R, Xv , Rv) of the pair (G, T) determines G as an abstract group whose presentation is given by chapters 9 and 10 of [Spr]. Here X and Xv are lattices in duality by the pairing (,), and R C X and Rv C Xv are dual root systems. Let o denote the ring of integers in F and p the maximal ideal of o. Let 7r be a fixed generator of p and q be the order of the residue field o/p. Let K be the subgroup G(o) of G; K is maximally compact. Fix a Borel subgroup B of G with unipotent radical U. Let R+ is the choice of positive roots in R that corresponds to this choice of Borel of group with R- denoting -R+. In general, for any positive system S of R, we write the corresponding unipotent group as U(S) and set U- = U(R-). Then U(S) is generated by unipotent elements u.,(t), where -y E S and t E F. Let D be the simple roots in R+ with D- = -D. The Haar measure m on G is normalized so that u 7 (o) has measure 1 for each y E R. Let 0 denote a C-valued character of U that is trivial on ua(o) but nontrivial on u,(p- 1 ) for each a E D. We also use 4 to denote a character of U~ that is trivial on ua(o) but nontrivial on ua(p 1 ) for each a E D-; this should not cause confusion. 13 Let T be the maximal torus in B. T is generated by elements p(t) for p E Xv and t E F. Say that p E Xv is dominant if (a, p) < 0 for all a E D. We denote A(7r) by 7rA. Now, we let C(XV) denote the vector space of rational functions in the indeterminates 7rcv", a E D. In C (Xv), we denote 7r& r O" by -rcv+Ov for a, / E R. We give C(Xv) a topology by completing with respect to the ideal generated by the 7r-av, a E D (that is, geometric series in 7r-&' are convergent). Let X : B -+ be the tautological character on B defined by X(7rA) = 7rA C(Xv) (caution: we take this notation for C(Xv) from [HKP] where it is used slightly differently). Let p be the halfsum of roots in R+. Let 6B the modular character of B. We have j1(2r") = q-(P). Let W be the Weyl group NG(T)/T with we the long element of W with respect to R+. The Weyl group acts on X and Xv, and for w E W we let Xw be the unramified character of B defined by Xw(7r")) = 7rw'"). W also acts on C(Xv) by w(7rP) = irw(p) and we extend this action by linearity. In addition, for a positive system, S set w(U(S)) = U(w(S)). We let we denote the long element of W with respect to R+. Let IndG(X) denote the principal series representation of G constructed by normalized induction: IndG(X) = {f : G -+ C(Xv)if(bg) = 6/ 2 X b)f(g) Vb E B,g E G}. The action p of G on IndG(X) is right-translation. Note that we do not require the functions f to be locally constant, for we will need to consider functions that are not locally constant everywhere on G. The Whittaker vector <5, is defined via the Bruhat decomposition G= U BnwU wEW 14 where the element n is defined as follows. For a E D, the element n0 is defined by nc,= u(1)u-a(-1)ua(1). Given a reduced decomposition w = sa...sam with ac E D possibly non-distinct, nw is the element nw = n, ... nam where n, is independent of the reduced decomposition. Definition 1.0.1 In Ind (X,) there is a vector called the "Whittaker vector" which we denote by #vo. We define it as a C(Xv)-valued function on G by 0",(bnwu) = /2Xv X (b)#v ,(nw)k(u) (1.2) where 6 OvOn)= and 6 is the Kronecker delta. We let # tV,wt denote The notation is meant to convey that the Whittaker vector transforms under righttranslation by u E U as OuP(g) = #u,"P(g)(u). We define the Whittaker functional W = W(U-, W(f) = When #u,Vp) f = k 4) by integration over U-: f(u)4-'(u) du. , the above integral is not absolutely convergent. We define W(p(g)- to be a limit of absolutely convergent integrals as explained in chapter 3. We 15 now can state the main theorem. Theorem 1.0.2 Let ao E T(o) and p E X' be dominant. Then -1/2 ~ W(p(aoir") - ) = (7r") w WEW (p_1 1q--R 17 ) . (-ER+ This document contains the proof of this theorem when R is a direct sum of root systems of types A, B, C, and D with some discussion of type E. The proof for exceptional types is forthcoming. The type dependence is solely contained in chapter 7.6. As mentioned above, this document will describe the similarities between the Bessel function and spherical function. We now summarize the proof in [Cas} of the formula for the spherical function. Recall that K is the subgroup G(o) of G. Definition 1.0.3 For v E W, let 0' E IndG(X') denote the normalized K-fixed (spherical) vector defined by #v (bk) = 6/2 v(b). We let OK denote #1. The Iwasawa decomposition G=BK shows that #K is well-defined. The spherical function F(g) is then defined by I (g) = K(kg) dk. ' 16 Compare our Theorem 1.0.2 to Theorem 4.2 of [Cas]. To establish this formula, Casselman makes use of the Iwahori subgroup I of K. Definition 1.0.4 Let S C R be a choice of positive roots. Define I(S) to be the subgroup of G generated by T(o) and the elements u,,(x) subject to the following conditions: x EP if 7 E -S x Eo if YE S. (1.3) We call such u,,(x) the unipotent generators of I(S). For our fixed positive system R+, we let I denote I(R+). The space of Iwahori-fixed vectors Ind3(X)' c IndG(x) has a basis (the "standard Iwahori-fixed basis") given by the set {#K-w}wDefinition 1.0.5 For v, w E W, define #vg., to be the vector in Indi (Xv) determined by #K;w(g) Let OKw denote = OK(g)1BnflI(g)- . Define the vector space VK(Xv) C Ind (Xv) to be the space spanned by the set {K;w}w; thus VK(Xv) = Ind((Xv)I. Casselman then introduces another basis of VK(X) which he constructs using the intertwining maps Tw : IndG(Xw) -+ Ind (X) (see chapter 5). These new basis vectors are {Tw(# )}w. Casselman shows that #K = OK Zw(TJ wEW in this basis is 7 yER+ 17 )TW('K l) (here we use the formulation of [HKP3). It remains to determine 1K TW (OK)(kir'") dk. Casselman proves that this integral evaluates to the simple monomial term, and the formula is proved. The formula in Theorem 1.0.2 was conjectured by the author after computations for certain G. Motivated by the similarity to the spherical function, we seek a proof of Theorem 1.0.2 analogous to that found in [Cas] for the spherical function. At each step of the proof in [Cas}, this thesis constructs corresponding objects in our proof for the Bessel function that play the same role. These objects include a previously unstudied subgroup of G which we call Iht and new vectors in the principal series representation we call Ou,,p;w (chapter 2), which take the place of the Iwahori subgroup and the Iwahori-fixed vectors, respectively. Due to the similarities surrounding these matrix coefficients, we call these two frameworks the K-fixed side and the (U, 4) side. We do encounter difficulties not present in the K-fixed side; to overcome them we develop new techniques, including novel ways to compute the Whittaker functional (chapter 4) and combinatorial objects we call paths in root systems (chapter 7). At this point, we do not have a foundational reason why the analogy between these matrix coefficients exists; searching for such a reason is one of many avenues of future work. Now we state relations in G used throughout the thesis. These are taken from the presentation of G given in chapter 9 of [Spr]. Fix a total order on R. Then, for linearly-independent a and # E R, there exist structure constants c7,;i,j 18 E F such that U.(X)uo(y) = uQ(y) ( uia+jO(cC,;ijXziyi) u0(x) (1.4) \ia+j,3ER;i,j>O where the product is taken according to our fixed order on R. We call this the commutator relation. We will make a slight abuse of notation where we allow the integers i or j to be negative and set Ca,O;i,_j = Ca,-/;i,j- There are also structure constants da,, E F satisfying nflu(t)na = us,.()(da,/t). Lemma 1.0.6 The structure constants may be chosen so that cQ,f;i,j E Z and d, E {±1}. Proof 9.5.3 in [Spr]. As da,O E {±1}, we will often suppress this constant during integration as it can be removed by a re-scaling of variables. We use the presentation of G given in chapter 9 of [Spr] where the field is F. For p EX, we have the relations p(x) u(t) p( 1 =W , and yv(x)n = uy(x)u 19 (-x'-)u^(x). (1.5) We will often make use of the following implications of (1.5). For xy y u_,(x)u,(y) = u7( + 1+ xy 1 1 + xy x 1+xy # -1, (1.6) and, for x # 0, u-7(x) = U,(-x~')-v'_-)nyu,(-x-1). 20 (1.7) Chapter 2 The Subgroup Iht and Vectors OU,b;w We define a subgroup of G called Iht. This subgroup will play the same role in evaluating the Bessel function as that of the Iwahori subgroup I in evaluating the spherical function. Definition 2.0.7 Let S C R be a choice of positive roots and let ht = ht(S) be the height function with respect to S. Recall that if 7 E S is 7 =mia1 +... + mrcr where ac are the simple roots of S and mi positive integers, then ht(7) = M1 + ... + M.r Define Iht(S) to be the subgroup of G generated by T(o) and the elements u-,(x) 21 subject to the following conditions: x E if Y E Pht (--y) -S (2.1) if Y E S. x E Plht (^) We call such u,(x) the unipotent generators of Iht(S). For our specific choice R+ of positive roots, we let Iht denote Iht(R+) and ht = ht(R+) when there is no possibility for confusion. Iht shares many properties of I. Indeed, I and Iht are isomorphic as groups (Lemma 2.0.12). A key property we will often use is the following. Lemma 2.0.8 Suppose ua(x) and ua(y) are unipotent generators of Iht- Then in the commutator relation Ua(x)uO(y) = U(y) Uia+Uj(cij;aOQXiY) Ua(x) each element in the product is also a unipotent generator of IhtProof Follows from a direct check of the cases according to the signs of a,3 and ia + j/3 along with the assumption that cij,. ,, are integers. El Lemmas 2.0.9, 2.0.10, and 2.0.11 are proved in exactly the same manner as they are proved for I. They may be proved by writing an element g E G as g = bnu for some b E B, w E W, and u E w- 1 (U-) n U. Write u as a product of unipotents, ordered by a convex ordering of w- 1 (R-) n R+. Starting from the right end of this 22 product, if u^, E I (or Iht), then apply a step of the explicit Bruhat decomposition (chapter 7.2) if u, V I (or Iht) and proceed to the next unipotent. Otherwise, if U-1 E I (or Iht), then move to the next unipotent. Then end result is an expression in BnYI (or BnyIht) for some y E W. We call this the explicit Iwahori (or Iht) decomposition. Lemma 2.0.9 Let S C R be a choice of positive roots. Suppose u is an element of the form JJ nyz u= 'YES where the product over S is taken in any order. If u E Iht(R+), then each u,(x,) is a unipotent generator of Iht(R+). Lemma 2.0.10 Let w E W. Then an element i e Iht has a unique factorization of the form i = i -(w)ai+(w) where i_(w) E (iht n w(U-)), a E T(o), and i+(w) E (iht n w(U)). Lemma 2.0.11 G is the disjoint union Bn,,Iht- G =U wEW More precisely, G= U Bn,,,(Iht n w-'(U-)) wEW and this decomposition is unique. Moreover, if g is of the form g = bnwu 23 with b E B, w E W, and u E w-'(U-), then g E Bnw(Iht n w-(U~)) if and only if u E 'ht- Let pV denote the half-sum of roots in RV+. Lemma 2.0.12 If pV E X', then Iht ~ nw, I (nwelrPv)1. Sometimes pV V RV+. In that case we extend the field F by adjoining 7ri/ 2 to obtain 2pv (71/2) which serves the same purpose as 7rP in the above lemma. Either way, Iht is isomorphic to I via an automorphism of G. Lemma 2.0.12 follows from fact (ii) of the following lemma. Fact (iii) will be needed in the proof of Theorem 5.0.31. Lemma 2.0.13 Let a be a simple root and -y any root in R. (i). (p, av) 1 (ii). (p, -yv) = ht(yv) (iii). - (3, -,v) = ht(yv) + ht(w(yv)) ,3EW-'(R+)flR+ Proof We make use of the invariance of (,) 24 under the Weyl group W. (i). Because sa(R+\a) = R+\a, we have ,P_ , v) =(s.P a, saV)) = (p - 2' -av ~aV = Therefore (p - 2, a') = 0 and (p, av) = -2 (c, av) (ii). This follows immediately from (i). (iii). The statement is equivalent to (-p-+ + 0 t = (P, W(-Yv)).- /3Ew-'(R+)nR+ But we check that -P+ E( = WR()p), ,3EW-1(R+)nR+ so the identity follows from the invariance of (,) under W. E Now we can define the "Iht-standard basis". Definition 2.0.14 For v, w E W, define 0",,. to be the vector in IndG(Xv) deter- mined by 0'pw(g) = #Ov, (9) 1 B.. I, (g)Define the vector space Vu,lp(Xv) C Ind (Xv) to be the space spanned by the for all w. Just as kK;w OK wGW 25 #, it is clear =UV > u,,O;w wEW by the Iht-decomposition for G (Lemma 2.0.11). 26 Chapter 3 Defining Integrals This section explains how we interpret the integral IU- Ou,,;w( ua) W 1(u) du that is not absolutely convergent. Recall that F has a topology induced by the non-archimdeadean metric and that FN has the product topology, for a positive integer N. Also recall that U- is a topological group whose topology is generated by the base of open sets MY (p") about the identity element, where y E R- and n > 0. Furthermore, U- is homeomorphic to FN as topological spaces, where N Definition 3.0.15 If f = IR+. is homeomorphism from some open subset 0 C FN into U- such that U- \ f(0) 27 f has Haar measure 0, then we call a parametrization of U-. An element of FN is described by an N-tuple (4i, ... , tN) of elements tj E F. Recall that a non-zero element t of F is of the form v7r", where v E o', n E Z, and ir is a fixed generator of p. The integer n is called the valuation of t and we call v the unit of t. Definition 3.0.16 Let f U- be a parametrization of U- . Given an : (FX)N -> N-tuple of integers 7 = (ni, ... , nN), define the subset f(ii) C U- to be the image w, f(oX7r"1 ... ,)o )7rN Let . denote a finite set of N-tuples of integers. Define the subset f(9) C U- to be U f ()= f(x) W1EJ Each set f(T) parametrizations f (ox)N. We require the to possess the following property. Definition 3.0.17 Let respect to Bni, and homeomorphic to is compact in U- f U-. We say that : (Fx )N U if whenever f(t 1 ,..., tN) is of the form aulnwIu2 28 f is torus-monomial with for some U1 , u 2 E U and a E T, then a is of the form a = a' Ma) aED where ma is some monomial in Q[ti, tl 1 , ... If a parametrization on f(..), f , tN, tN1 is torus-monomial, then the Whittaker vector # for a fixed -/, in the sense that the unramified character only finitely many values. As is bounded J / 2 x takes on f(.0) is compact, it follows that /f P) U4(ua)@ -1(u) du is absolutely convergent for for any a E T. Likewise the corresponding integrals for p(a) - #v,0- are also absolutely convergent. Definition 3.0.18 Consider a sequence sets of N-tuples {fA}k. We say that - oo if for any positive integer M, there exists some ko such that for all k > ko, _fk contains each N-tuple (n 1 , ... , nN) satisfying nil < M for 1 < i < N. For a torus-monomial parametrization f W(p(a) - #U,;w) = lim k--+oo We will present, for each and a sequence f, k -* oo, define .. (ua)O-1(u) du. k #U,;w, torus-monomial parametrizations f that depend on a reduced word for we. We show that the above limit exists and is independent of the choice of reduced word. The parametrizations are constructed in chapters 4 and 7.2. 29 We then define the Bessel function Wup(a) to be Wu,1p(a) = E W(p(a) -#vp-.)WEW 30 Chapter 4 Computing W(p(a) - <p ) via a Quotient Space The main result of this section is Lemma 4.0.23 which partially constructs the torusmonomial parametrization used to define W(p(a)- Ou,';); the remaining construction is in chapter 7. In this parametrization, Lemma 4.0.23 expresses W(p(a) - OU,.,) as a product of two factors. This expression is central to the proof of Theorem 5.0.31. Recall that W is defined as an integral over U-. We break up this integral according to a subgroup H of U- and its quotient space. We state the following theorem. Theorem 4.0.19 Let G be a locally compact group and H a closed subgroup. Then there exists a G-quasi-invariantmeasure IG/H on the quotient space GH and j f(g)dp(g) = J (IH f(gh)dPH) dPGH(gH). Proof Chapter 1 of [OV]. 31 1 For a w E W, we apply this theorem with G = U- and H = w- (U-) n U-. The following result characterizes certain coset representatives of U-/(w-'(U-) n U-) we need for Lemma 4.0.23. Lemma 4.0.20 UT(w 1(U-) n U- n Iht) U~ n (BnwIht) = where the disjoint union is indexed over the elements Ti E U- n (Bnw(w-1 (U-) n U n Iht)). That is, we may assume -9 are coset representatives of the form Ti = bnwi E Ufor b E B and i E w- 1 (U~) n U n Iht. Proof Follows from the Lemmas 2.0.9 and 2.0.11. Lemma 4.0.21 Let sn.. .s, be a reduced word for w E W, where si is the simple reflection for ai E D. The representative7U of U-/(w-1(U-) nU-) may be written as n Ti u-ai (ti). = i=1 Furthermore, when U is of the form auni with a E T, u then a is of the form H aV(M) acrD where ma is a monomial in ti,tj1 , ... ,tn,tj. 32 E U, and i E w- 1 (U-) n U, Proof For a fixed w E W, almost every element u of U- may be written as HU-ai) \i=1 (4.1) H / (-Ew~l(R~)nR~ This may be proved by writing unrI nwe as ulnwU2 for some u 1 , u 2 E U; this can be done for almost every u E U-. Then we may write U = 'iinw t'2/Ui-(ai)(ti) for some u'E U- n w- 1 (U-) and where wi = si...s1 . This in turn may be written as U1' naiUai(ti) n2- Now, starting with i = n and proceeding to i = 1, apply the relation naua,(t) = uac (t-))a(t)u~(t-1) and move ua1 (t-1) to the left as in Lemma 7.2.1. The resulting expression for u is of 33 the form u = bu' with b E B and u' E U- of the form (4.1). Thus b = 1, proving the statement. To see that a is a monomial in the ti, t;-1 , proceed from i = n to i = 1, applying the relation u-C,(t) = uoj(t-)av(t-1 )n-uc(t 1) to U and move ua,(t~') to the left using the relation ua(x)ua(y) = U-a(y(1 + xy)~1)aV(1 + xy))ua(x(1 + xy)- 1 ). Then by induction it follows that the parameters of a are monomials. I We now define two terms that arise from the quotient integral of Lemma 4.0.23. Definition 4.0.22 Let a = ao7r" with ao E T and p G X' dominant. WH(XV, Define w) and WU-/H(Xvwa) by WH(XIW) = w-'(U-)nU- ov~,v1 ;w(nwu) du and WU-/H(X, w, a) = 6-1/2XV(a) 1/2 Ih v(b)V-'(a-ua-1)_I n.rs )dU. where T! = bnwi according to Lemma 4.0.20 and du- is an abbreviation for the quotient space measure dp -/w-1(U-)nU-(U(w- 1 (U) n U-)). The torus-monomial parametrization defining Wu-/H(xv, w, a) is constructed in the proof of Lemma 4.0.23 and in chapter 7.2. 34 Lemma 4.0.23 Suppose that a = a01r" with ao E T(o)/T and [ E X' dominant. Then W(p(a) = WH (X, w)WU-1H(X, w, a) (4.2) Proof We have 1 J/ U -U#j,;.(a)W (u) du = J-1/ 2 X (a) j u-U 1 .. (u)W- 1 (aua- ) du. conjugating out from a change of variable in u. Apply Theorem 4.0.19 with H = w-1(U-) n U- to get j (a) i/2Xv fU-1H By the support of # JH #OU,0w(Uh)0-1 (a-a-')O-~ (aha-1) dh du. U we may assume that u = -Uh E BnwIht. Lemma 4.0.20 then says we may assume h E HnlIht and U = bni with i E w- 1 (U~)nUnIht. Therefore h lies in the kernel of 4, and the dominance of a implies that aha- 1 does as well. We obtain 6112 Xv(a) 6x(b>#"j,,(nihp1(aua1)dh di. H JU-1H JH Next we move the unipotent generators that comprise i to the right over h. We refer to Lemma 7.2.1 about moving unipotent elements. Make changes in h to cancel the effect of these moves; these changes take the form of the operator B. which is discussed in chapter 7.2. Since each new unipotent must be a unipotent generator (Lemma 2.0.8), these changes do not alter the fact that h ranges over H n Iht or that aha-1 lies in the kernel of 4. 35 We finally obtain 1/2Xv (a) f-H J H, 1 )1IBfwIh H/X(b)4,;w (nh)1-'(a7a- (U)dh du which factors into the two integrals WH(Xv, w) WU-/H(Xvw, a). L Theorem 7.1.1 gives the following important result needed in chapter 5: WH(X , W) = -7r-w(-')qht(w(y))-ht(y)-1 I- 1 7yEw- (R+)nR+ 36 (4.3) Chapter 5 Intertwining Operators Applied to oU,0;W There exists a unique intertwining map T,, : Ind (X.a5v) - IndB(X,). It may be E IndB(Xs""). For the purposes of this expressed via an integral for appropriate document, we will let T, denote the integral itself. Definition 5.0.24 For a E D and # a function on G, define T,, by f (noau(x)g) dx. lim Ts.(#)(g) = m-oo J,-m Remark 5.0.25 In the K-fixed side, the intertwining maps T, are used to construct the vectors T,(q ) for each w E W. These vectors are in fact a basis of VK(X) and are instrumental in the proof of the formula for the spherical function. Therefore, we would like to consider the vectors T,(#w"73)) in hopes that they constitute a basis of Vu,,p(X). Upon computation, however, it turns out that these vectors may not even lie in Vu,,(X). Nevertheless, we can still obtain a critical result "modulo 37 the kernel of W: we consider a multiple integral determined by W o T,, over the IR++I 1-dimensional set nruaU~. The Ts, thus aid in establishing relations among the Wu(,#;). To explain this result (Theorem 5.0.31), we first define actual maps between the Vu,,,(X). Definition 5.0.26 For -y E R, define cw(-y) E C(Xv) to be 1 - q1r7V 1- 7r--tv Definition 5.0.27 For w E W and a E D such that e(saw) = 1 + f(w), define the map TI', : VuO (xso) T.'a(#, S.) T'0 = =-r -+ VU, (X') by (cw(v- 1 (a)) + 7v1(av)) kV,;w - q "vU)#,p;W + (cW(v- 1 (-Y)) 7r #V(V)O$ (5.1) + q(5.2) 5 define T', by For a reduced word of w = s ... s1, T' = T,', 0 ... a T'. Lemma 5.0.28 The maps T' are independent of the reduced word of w used to define them. Proof The T' are essentially multiples of the maps T,, fixed case; see Theorem 3.4 of [Cas]. applied to VK in the K- These maps on VK are independent of the reduced word for w because the intertwining integrals are themselves independent of the reduced word. The same independence follows for the T'. The definitions (5.1), (5.2) then determine T' on all of Vu,,(xv). 38 Recall that Gindikin-Karpelevich formula for the K-fixed vector states that the takes the K-fixed vector 0'a" to a multiple of intertwining map T, #' ; there is analogous result for the Whittaker vector. Theorem 5.0.29 For a E D, TS(#"gV) = T ,' =W(V-(a))#,. Proof Recall that the Whittaker vector #U E IndG(xv) is supported on the double coset Bn,,,U where it is evaluated as #jp(bnvu) _ e12XV(b)b(u). We may assume that g bnrmu for some b E B, w E W, and u E U. After a change = in x, we get lim 6 2 xv (b) f q c" (nau,(x)niwu) dx for some c, d E F. Consider three cases for w. (i). w = we. We have #v (bwu) = 6112 xv(b)(u). In the argument of Os', apply the relation nu,(x) = u (x-1 )av (X-')u -(a 1 ) to the left over u-,(x- 1 ) by conjugation to obtain and move the n, 6 2XV (b) j #OL(ua(x- 1 )av (x- 1 )nwu-w(o)(x-1 )u). 39 Since -wt(a) is a positive simple root, the integral reduces to i xv(b)V)(u) IF 612 xsV v (-1))V(x-1) dx. Letting m -- oo, we get cW(v-1(y))J1/2 Xv(b)*(u), verifying the theorem in this case. (ii). w = sawj. Then #rY,(bnu) = 0, and jp-m $s (no,(x)bna nw,u) dx = 00' (riwt uwe(a) (da,we x) u) dx IF = 1/2 Xv(b) &u) j 0(da,,Wx) dx =0 for sufficiently large m. (iii). w 5 we, scwj. Then 0yo(nw) = 0. The integral in this case is also 0 because nau-a(x)nw is not in BnwU for any value of x, and the final case is proved. The second equality easily follows from the definition of T,', and by pairing up w with saw for each w E W.0 This formula will be used to construct the normalized intertwining maps Tb'. Ultimately we show that the set of vectors Tw(# -1 1) is the desired basis of Vu,,P(x). Definition 5.0.30 Define TZ: Indo(xSaV) -- IndG(Xv) by = T 40 For a reduced word of sa...si of w E W, define Tw : Ind (Xw'") - IndG(xv) by Tw = Tw o ... o Tw. We check T8Vi(#O'(q) T o Tw(#vg.) =O#", (5.3) =". for all w E W. Equation (5.3) implies Two Tw = Tw W1 W2 W1W2 for all w 1 , w 2 E W. These facts will help us express (5.4) # in terms of the T,(Ow"'.). We emphasize that the purpose of the intertwining maps in this paper is to establish relations among W(p(a) - .V). The intertwining maps serve the same purpose in [Cas] and [HKP] in the K-fixed side. Casselman first describes the intertwining maps on VK(x) by first directly computing the intertwining integral for T8,,(#K;1)He then obtains the result for arbitrary w by applying the action of the IwahoriHecke algebra (see his proof of Theorem 3.4 in [Cas]). At this point, we do not have recourse to such a tool for the (U, 4) side. Therefore we prove the analogous result outright for general w. This is the content of the the next theorem. Theorem 5.0.31 Let a = a0 irA where ao E T(o) and A E X' is dominant. Let a E D and v E W such that f(sav) = 1 + £(v) and let w be any element of W. Then cwPV(V- (-y))W(p(a) - 0'",w) = W(p(a) - Ts,("Q;)) 41 Proof Consider the Whittaker functional applied to the intertwining integral W o Ts, (p(a) - J J4 w(naua(x)u)-(u) dx du. We will construct a parametrization of this domain of integration that is torusmonomial to evaluate the integral as a convergent limit. Then we will evaluate this integral in two different ways which will give the two sides of the equation in the theorem. To construct the parametrization, we begin with the parametrization u 0 (x)7h, where TUh is the parametrization of U- from chapter 4. We then perform changes of variable to this expression to create a parametrization that is torus-monomial. First conjugating a to the left and then making a change in the u and x gives B l/ 2 (a)Xv(a) J # ,;w (nou.(x)u)-'(aua-1 ) dx du. Recall U U= n Bn,Iht yEW and that the set ncaUa(x)BnyIht c Bn.Iht only if y = saw or y = w, depending on x. We may therefore restrict the variable u to Bna.Iht and BnriIht. We divide the proof according to whether f(saw) = 1 + f(w) or f(saw) = -1+ f(w). f(saw) = 1+ e(w) : u E Bn. ,,Iht -/2()" : Consider j j. IF 1 U,(nuo,(x)u)P 1 (aua- ) dx 42 du. Since f(saw) = 1 + i(w), we have (saw)- 1 (a) E R-. Write u as Tih as in chapter 4, with h E H = (sw)'1(U-) n U- and give -th the torus-monomial parametrization. Express Uh = bnrni. Move b to the left over ncua(x) and make a change in x to reverse the effect of this move. As in the proof of Lemma 4.0.23, we get (a)X (a) SB J UnBn ,,w Iht 6 1/2 Xv (b) B j F Re-arranging, we get in the argument of ky-(1)aua(x)nan h)V U~ n Iht- dx du. #"y" (5.5) nwuw-l(-a)(x)h. That w 1 (aa O;noaxcwhO1(fu (-a)E R~ implies x E Pht(w'(-a)). Now uw-1(_,)(x)h ranges over w- 1 (U-)n Make the change in x so that (5.5) so that has the torus-monomial parametrization. Integrating with respect to x and h gives WH (XsV w) and with respect to U gives WU-1H (X, sow, a). Now (4.3) implies WH (X S, WH (XV w) r qM1 sw) q sow(-Yv) + E where '7ER= 'YER+ _1 43 sc~w (-yV) 1 al and (htsaw(7v) -ht 7). - (ht w(yv) -ht-y) M1 = 'YERt 'yER_ We simplify A, and M 1 . The fact R' , 1 = R _w, \w- 1 (a) shows that A, =av We claim M 1 = 0. First we collect the terms involving roots and immediately see they reduce to -ht w- 1 (a). Re-expressing the sum of coroots using w(Ri ) -R gives -htav+ E ht-yv-hts.(yv). 'fERtwt This expression reduces to (a,V) -1+ 7YERWjive which is equal to ht w- 1 (a) by Lemma 2.0.13. Thus M 1 = 0. Putting together these results with Lemma 4.0.23, we see the contribution from integrating over these regions is vi W(p(a)- vsc- U q u E Bn.Iht We now express u as 7h, with h E H = w- 1 (U~) n U-. By reasoning similar to that of the previous case, we consider 6l/ 2 (a)xv(a) B ~ JU-nBn,,Iht 6 1/2xv(b) Sev j B F (nuo (x)nwh)- U Apply the relation U-I(x)nc = Ua(x 1 )av (x-1 )u 44 -(x 1 ) 1(aUa- 1 )dxdu. and re-arrange to obtain 6 61/2 Xv(b) B 1/2(a)XV(a) 5 1/2 Xscv(av (X-1))(NU,,Ow(nwu-w-l(c,)(x- SU -nBn,, Iht That -w- 1 (a) E R- implies x E pht(w 1(a)). Make a change in h n wg-is o h w-(e)h. Integrating with respect to x and h gives ht(w-1(C,)) -raV ( (1 - q-1)V-1 WH (Xs v) IF 1 IF-- while with respect to I gives WPU- 1H(x, w, a). Now H (Xav, W)) WH(Xv, W) v-'(A) where A2 = -sa(w(yv)) 1 YER+W'tI_ Re-express this sum as before to get (a, V)aV -YERZ.t 45 + W(.Yv). )h). which is (1 + ht w~ (a))av by Lemma 2.0.13. Putting together these results yields cw(v-'(y)) + 7r" (C)W(p(a) - #vUP;w) The two cases together show the total contribution is (ca(Xv) + 7rv-1(a))W(p(a) - Ov;.) - q- 17r -(av)W(p(a) - #,;) which is W(p(a) -T,'/(#7";.)). e(saw) = -1+ £(w) u E Bn,,..Iht By the reasoning above, we consider 6J 1 / 2 x(a) JUflBn.scwIht That -w- 1 2 x(b) j #yP# ,(naua(x (a) E R+ implies x (E p1-ht(w-'(a)). newh)O(aia-1) dx du. (5.6) The parametrization of h is of the form h = UW-1(a)(t)h' (5.7) with t E phtw-(a) and h' a parametrization of w-'(U-) n U-. Apply to ncaUa(x)u-a(t)nriwh' the relation ua(x)u-.(t) = U-( t x + f)av ((1 + xt)-)u( 1 x I+ xt 1+xt 46 (5.8) and the fact that 1 + xtw-l(,) E o' allows us to write the argument as U,( using the left U-invariance of in the limit f #P x X )nwh'. 1 + xt The integrand is now independent of t which, -+ oo, ranges over pht(w-1(a)). Applying the explicit Bruhat decompo- sition to h' yields __a I7r Ariwtu U-a( for some 7rA E X' X 1 + xt )0orawu From chapter 7, we may assume that each and a' E T(o). parameter t^, of h' has valuation ht(-y). This, along with x E p1-ht(w 1 (a)), implies that ir-Au0( X 1 + xt E U-(o). )irA is then independent of x as This implies, after one more re-arranging, that well. Thus the integral is absolutely convergent. Integrating over x, t, h' and -9and letting m, -+ oo yields, respectively, m(p1-htw-1(a)M(phtw1(a))WH (Xsv, ) WU-/H (Xv , sw , a). Combining 1 1 m(1-htw (a))M (phtw (a)) _ and the previous calculation WH (X ",CW)) qirv-l(av) WH (XV, SaW) 47 -1 gives us the contribution '1'v)W(p(a) - #Ov.) -7r"V u E BnwIht By the reasoning above, we consider J1/ 2 x(a) flBnwIht l/ 2 x(b) j U;w(ncua()nwh))(auia-1 ) dx du. (5.9) The relation E pl-ht(w-l(a)), as -w-'(0) E R+, yielding implies x- j-1/2 -1) - nnur(X) = ua(x-')av ()x (a 2 1/2 X/ /U-nBn.,.It .B v () F BX (- Xsav ( 1 1 )-( (x1)uwu-(x'nh. The same argument of the previous case shows that #" is independent of x. Integrating as before gives (1 - q-1)V-1 1-7r-cV S WH (X sav,7 )WU- /H (Xv, w, a), as in the case f(saw) = 1 + f(w), u E BnwIht. This expression is then (cw(v-1(y)) + q~ r"v(av)) W(p(a) - #"y,,) 48 The two cases together show W(p(a)#u,.p;w) + (c', + W(p(a)Ts. (#u,,;s.w)) = -7r' )W(p(a)#u,,;sw) which is W(p(a) - T',(0'-'.w)). We now perform the same integration in a different order to obtain the right side of the equation in the theorem statement. For # E IndG (X), write W(T,, j j (0)) as the double integral 1 du. #(u-a(x)nau)V- (u)dx The relation 1 u-a(x)na = uc,(x-)av (x-1 )u-(x ) and the change of variable u - U_a(-X- U yields (F/ 61/ 2 xV(av (X-1))-1 (u,(x-1) dx) (I - q#(u) 1 (u) du). The integral with respect to x equal to cw(v- 1 (a)) and with respect to u is W(O). Corollary 5.0.32 For w 1 , w 2 E W and a as in the lemma, -1 -1 W(p(a) - Tj(#w)) = W(p(a) - Proof This follows from straightforward induction. 49 'U -2- 50 Chapter 6 Evaluation of the Bessel Function This chapter evaluates the Bessel function Wu,,p(a) at torus elements a of dominant coweight. Lemma 6.0.33 expresses the Whittaker in the basis T,'1,(O'7 1 ), w E W. We use Corollary 7.1.2 from chapter 7. The arguments in this section closely follow those in [HKP]. It follows from the definition of T.' that T,(#5~. 1) is a linear combination 1= v<w such that the coefficient a, a is nonzero. In particular, 17 -cw(Y)- = = q- 7r. (6.1) YEw(R-)nR+ Therefore, there is some triangular, invertible matrix which expresses the change between the basis the Let's express #g,, # 1 .. and the T ( in this basis. 51 Lemma 6.0.33 W( qER+-1 'YER+ wEW Proof Set .) d. #g= (6.2) wEW for some constants dw. Recall =U1 Z U4,;w. wEW Equating the coefficients of #U, .;w, in both expressions yields dwaww, = 1. Therefore I-cw(y)q7r~v dwe = - 'ER+ 'YER+ S1-q7r~,v 1 7r-,Yv Now the dw satisfy dw1W2 = W1(dW2)- To see this, apply T , to (6.2) and then apply the Weyl group action w, to recover vectors in Vu (X). From the expression for dwe, it follows 1- dw = w(f 1 YER+ 52 qiryV 7 V_ ). by taking w 2 = W, W1 = wfW-w. n Theorem 6.0.34 Let a = ao7r with a E T/T(o) and W(p(a) - Ub) = 61/2 (") E X' be dominant. Then ZW(17J -JER+ WEW Proof Apply p(a) to the expression for 7V )Y # in Lemma 6.0.33. Wo T', = W implies w( WV(p(a) - #7,3) wEW l 1- 1- q7rYv 1r7y )W(p(a) -YER+ By Theorem 7.1.1 WY(p(a) - #',11) = 61/2(ir-')W(irj _1_yv7 YER+ These results together provide the formula. L 53 1 The fact that 54 Chapter 7 Computing WH(X, w) 7.1 Statement of Theorem Recall that WH(X, w) is defined by WH(X,W) = Jw-'(U-)nU- U,; du. (7.1) This chapter is devoted to proving the following theorem, which was used in chapter 5, and its corollary, which was used in chapter 6. Theorem 7.1.1 -rV WH (XW)= ht(w(yv))-ht(y)- -Yw(R+)nR+ Corollary 7.1.2 W(qdUp;1 =vJ~ 1 7 PER+ Proof Apply the theorem with w = 1 and use the support of 55 .1,p~ plus the fact that ht (7v) = ht(-y) YER+ 'YER+ for any root system R. Our approach to proving Theorem 7.1.1 consists of the following steps. We first parametrize w 1 (U-) n U~ using unipotent root elements ordered by a convex or- dering of the roots in w 1 (R-) n R-. After an "explicit Bruhat decomposition" in section 7.2, we express elements of n,(w 1 (U) n U- as the product n4 u+, with u+, u' E U and a E T. We make this decomposition because the definition of the Whittaker vector requires its argument to be in this factored form. This Bruhat decomposition then determines a sequence of changes of variable that simplify the torus element a. Under this change, X(a) depends on only the valuations of the integration variables. These changes determine the torus-monomial parametrization. We thus partition the domain of integration into cells for which each integration variable has a fixed valuation (each cell is of the form f(iT) in the notation of chapter 3). Integration over such a cell then becomes an integral over a product of unit groups 0 '. We show that this integral is zero for every cell except one. To perform the integration over each cell, we introduce in section 7.4 coordinates in the unit variables which we call "nesting coordinates" that make the integrand easier to handle. In particular, they isolate sub-integrals of the following forms: / 0, n > 1 = =(Ur-n)du j/ (z7r-1) dz = 0 (7.2) (7.3) These are are two basic ways to prove an integral of V) vanishes. Section 7.6 identifies key terms in the integrand with combinatorial objects in the root system R. After 56 analyzing these objects, we can finally carry out the integration in section 7.5. 7.2 Explicit Bruhat Decomposition Let u be an element of U-. Let N be the the number of roots in R-. After a sequence of N steps, we express u as u'NaNntWIUN with UN, u' E U and aN c T, provided that u is sufficiently generic. Any element of U- can be written as U = 7 UY(ty) (7.4) yER- for some t- E F. In general, the roots 'y may be taken in any order. For our purposes, we always choose convex orderings; the convex ordering ensures that, among other things, the group operations we apply are well-defined and will eventually terminate. Recall that a convex ordering of the roots in R- is equivalent to a reduced decomposition we by the following correspondence. Let N be the number of roots in R- and let {aN, ---, 2, a I be a sequence of (possibly non-distinct) simple roots in D+ that corresponds to a reduced decomposition of we: We = SQN'.Sa 2 S'1' Define wj by w= sa ...so, with wo = 1. For i > 1, define -yj by Wi= i(ai) 57 Accordingly, we say that 'yi maps to the simple root ac. The ordering determined by yi+1 -< -yj is a convex ordering of R-. We will achieve the Bruhat decomposition U= via a sequence of steps. Our initial expression for u is (7.4). NaNnW-1guN N After the jth step the expression for u appears as U =uajnwi1 Y)(fyj) Uj (7.5) where j = +1s,( J7), i=j aj = Aj(-1) J7yv (fI), i=1 uj is some element of U, A3 is some element of Xv, and fy,j rational expressions. The expression (7.5) reduces to (7.4) when the final form when j E F(tyl, j ... , ty) are = 0, and attains = N. We show what each step of the Bruhat decomposition entails. Assume that we have expression (7.5) after the jth step. As -wj(yj+1) is the simple root aj, apply relation (1.7) to u-,g, in the product over roots 7 -< 7j to obtain 11 um(-Y)(f-Y,i) The rightmost factor uj+ 1 (fN+) uaj+(f .Ql~)a, (+)n a ( ) (.) in (7.6) will remain in this position and contribute to uj+1. The other unipotent factor ucj 1 (fj+ 1 ), however, will contribute to u . Therefore we have to make sure that we can "move" this unipotent to the left; Corollary (7.2.2) provides this result. We postpone the proof and assume for now 58 that uC,+ (fyi+,)aY+1 (f-i7,ij)" H uwj (Y)(fY-,j) 747j+1 is equal to u'ca,+(fj+i)na3 for some rational functions (I W( 1yj+ -<~Y (7.7) 1 (y j+1 (f i+ fy,j+1 and u' E wj(U). Therefore this u' may join the unipotent element u and we set u+1 to be Uil= ujajnw,3 u ri, a3 in addition to setting aY1(f aj+1 = =+Wa-n Wi± )n and Uj+1 = uajy (f -~~)uj. This completes the j + 1-th of the Bruhat decomposition. The following technical lemma about moving unipotents in general is needed to prove the result assumed above. Lemma 7.2.1 Let 'yN -< ... --< y1 be a convex ordering of S C R. Suppose we have a product consisting of u-, in this order with other unipotent elements uO = udim interspersed between them: U-7N 'YM-1 ... 7-n+1 U-Y )On,1U)3,2''... 59 n -1,1 00n-1,2 ... 'U,1,1 O1,2-''' 71 (7.8) The 1,m may be any roots in R subject to the following conditions: (i) If #i,m E S, then (ii) If E S, m E, #i,m = yj for some -yj -< -yj. then 3i,m = -7j for some -y >- -y. Then we may re-write the above product as 1 u' 1 u, (7.9) i=N for some u' E U(S~) and with parameters of the u, possibly altered. Proof We obtain the expression (7.9) by sequentially "moving" the uf3 "to the left", "passing over" the other elements uy in the product. We show each move must end in one of two ways: either we reach the far left end of the product, where the uo contributes to the u' factor; or we reach a u^, with -yj = 3, in which case the uo is absorbed into uny by the relation u, ()u0(y) = U, (X + y), (7.10) for whatever parameters x, y that these unipotent elements have. It is in this manner that the parameters of the u, may be altered, as referred to in the lemma. The element we choose to move will always be the "leftmost" of the u,, u,3, 1 in particular in (7.8). Of course, during this move we may create "new" unipotents by the commutator relations, which we label as uI3 ,, appropriately. We will show that these new unipotents also satisfy conditions (i) and (ii). Thus, we show that conditions (i) and (ii) on products in the same form as (7.8) are invariant under moving the 60 leftmost u3 to the left. First, suppose 3 , , G S. Then by condition (i), to the left and absorb it into uYi as in (7.10). 3 n, = -yj -< -y,. We move u, Now, if we have 7i between -yj and y, such that 7j and 7 span an irreducible rank 2 root system within R-, then u,, creates new unipotents as it moves to the left over uy, with roots in this rank 2 system, appearing in a block immediately adjacent to u- on its left. The convex property of the ordering implies that each of these roots must be equal to some with 7y -< 7y -< -y. 'Yk Thus new unipotents of this move satisfy condition (i) (in this case, any new unipotent must lie in S so condition (ii) does not apply). On the other hand, suppose 3 n,,1 E S-, so -1, = yj >- -7 . Then we move up,3r to the left end of the product. Condition (ii) ensures that this move does not create torus elements by relation (1.6). Now, if we have -yj -< y with -- y and 7fj spanning an irreducible rank 2 root system of R, then u3, creates new unipotents as it moves to the left over u . To see that the roots in this rank 2 system satisfy conditions (i) and (ii), consider the sequence {-Y5, ... , -72, -71, 7N, 7N-1, - ,7j+1} which is a convexly-ordered positive system. This proves the invariance of (i) and (ii) under moving uo, 1 to the left. Now, we claim that iteratively moving the leftmost uo to the left will eventually yield a product in which no uf3 appear, that is, a product of the form (7.9). This is proved by induction on the position of the leftmost unipotent u3 1 . with n the base case. = N as l Corollary 7.2.2 In the previous notation for the explicit Bruhat decomposition, the expression Ua S(f-Y) 61 (f-) (7.11) may be re-written as (7.12) H U-(f-y,j+1) ' for some u' E wj(U) and fyi,j+1 E F(tY..., ti) Proof Apply the lemma with S = wj(R-). E The explicit Bruhat decomposition shows how to express u E U- as u'NaNWfUN. Recall, however, the argument of the Whittaker vector in (7.1) is n.u for u w- 1 j = (U-). We can view nu £(w) and fy,j E U- n as a j-th step in an explicit Bruhat decomposition, with = ty. Thus we can use the same procedure to express neu in the form u'NaNnwe UN. We therefore assume w = 1 from now on, as the general case translates easily from this one. Using 0 uaj+1 ~ UN 1 1(f ,j), (7.13) j=N-1 we write that integral for W(Ou,+;i) as fU- j#,.;i(u) du = fU- X(aN) (f\<j<N 1BIht (u) du. /y~~ (7.14) We now describe changes of variable in the t, that render the rational expressions fY,,j into monomials. Changes of Variable From Moving u-. Given -y E R- and a convex order -< on R- , consider the product ( Y,(tY,) \s'-<y/ 62 U_7(t-1). Move the unipotent u-, and then all new unipotents u3 to the left as described in Lemma 7.2.1. The u, may get absorbed by a u, via usi(t 7 )uO(x) = uY (t, + x), where #3 = -yj and x is some polynomial in the parameters. In this case, we make the change of variable t, ++ ty - X. The order of these changes is determined by when they occur in moving the new unipotents to the left by always moving the leftmost u,3 . We call the set of all changes of variable that occur in this manner the changes of variable from moving un. We denote the application of these changes by the operator By. We also let B>. 7 and B-.<y denote the changes of variable from moving uy, for -y' - y and -y' -< -y, respectively. The order of these changes is determined by the convex ordering, starting from the most succeeding root. The B signifies that the changes are derived from the explicit Bruhat decomposition. The B-<, simplify the elements aN and UN so that there parameters are mono- mials in t,. The next lemma describes these monomials. Lemma 7.2.3 In the notation of the explicit Bruhat decomposition, let m,, denote B>-y, (f,,j) and my, In addition, Bsy 1(fyj+,1) to denote m,,,. = By,(fy, Then ). Proof Follows from the explicit Bruhat decomposition. 63 On the other hand, the changes B-,y, complicate the support condition 1 BIht (u) in the integrand of (7.14). Namely, in the original coordinates (7.4) of u, the statement u E BIht is equivalent to (7.15) t^ E phty for all -y E R-. After the changes of variable from moving uy, the conditions (7.15) appear as E pht" We now partition the domain of integration FN into cells on which B y1 (x(aN)) is constant. Recall FN is parametrized by t = t71 I (t7N7I' 7y E F. An element t E F is of the form t = v7r- where u E oX and n E Z is v(t), the valuation of t. Definition 7.2.4 Let T denote an N-tuple of integers 7 = (nN, ---, n1). Call the subset of FN such that v(t.,) = ni for each i the cell C(ff). ty, = On C(), v,7rni with ni fixed and v., free to range over o'. Since x is unramified, X(B-<),l (aN)) depends only on 7T: let x(7T) denote the value of X(B, (aN)) on the cell C(T). Clearly FN is the disjoint union of C(T) as E ranges 64 over all N-tuples. Therefore we consider the integral I(T) over C(7T): I(T) = X(7) j (( m-1) Jcn)y<O / (7.16) dt,. 1pht,(Byl (t-)) Y<O ) -Y<O We will show that this integral is 0 for every 7 except one, namely 7E = (ht(-YN), -. , ht(-yi)). Paths in a Root System 7.3 We first examine the structure of these functions by defining combinatorial objects which we call "paths in a root system". This definition is derived from moving the unipotent element u-, to the left and thus will be used to describe the functions BO(t,) Definition 7.3.1 Let S C R be a positive system. Let P denote the 3-tuple of the following sequences. For an integer f(P) > 0, let {ae(p), ..., a,} and {be(p), ... , b} be sequences of integers in {±1, +2, ±3} and {P(p), ... , P 1, PO} a sequence of roots in S. Define P,i to be the subsequence {P, ..., P}. Define E(P 1,O) to be a 1P 1 + biPo and, for 1 < k < f(P), define E(Pk,o) to be E(P,o) = akPk + bkY(Pk-1,o). If E(P,o) E R for all 1 < k < £(P), and if b1 < 0 with all other ai, bi > 0, then say P is a path in S. Furthermore, given a convex ordering -< of S, if E(P) -< Pt(p) and Pai -< P for 0 < i < e(P), then say P respects the convex ordering -< of S. 65 Lemma 7.3.2 Let P be a path in R- that respects the convex ordering. Then for k > 0, E(P,o) -< E(P+1,o)- Proof Follows from convex ordering. Definition 7.3.3 For a path P in R, define c(P) E Z to be f(p)-1 c(P) = Jl CessiE(Po);ajba . i=O Furthermore, define t(P1,o) to be t(p1,o) = taltbi and t(P,o) to be t(Pk,o) = takt(Pk_,o)k with t(P) = t(Pe(p),O). Remark 7.3.4 With these definitions, we can characterize the elements in each change of variable from moving u-po. Each change is of the form tE(p) H-+ tZ(p) - c(P)t(P) where P is some path that respects -< of R- beginning at P and which sums to E(P). Definition 7.3.5 Let P and Q be two paths in R. We say that off P if Qo = Po and E(Q) = P for some i 66 $ Q E T branches 0. We call such a pair of paths (P,Q) a branching pair, and that P is the absorbing root for this branching (as this root corresponds to a unipotent absorbing another unipotent). We give the following rooted tree structure on paths in R. Let T be a rooted tree and label the vertices of T by paths in R-, not necessarily distinct. We call T a rooted tree of paths in R- if the following property holds for each vertex v of T: the path of a child vertex of v branches off the path P of v such that the paths of the child vertices have distinct absorbing roots in P. Definition 7.3.6 Let (P,Q) be a branching pair of paths in R. Let k be the greatest index such that P = Qj for all i < k. We say that P -<Q if Pk+1 - Qk+1, Q -<P if Qk+1 < P+1- Definition 7.3.7 For a path P in R-, define the operator Bp inductively by Bp(ty) = >7 t, - t(P -< Q)c(Q)BQ(t(Q)) E(Q)=Y, Qo=P and extend this action to Q(t,, ... , tYN* Lemma 7.3.8 BO(t,) = t, - Z E(P)=y, 67 c(P)Bp(t(P)). Po=0 7.4 Nesting Coordinates The integral (7.16) is actually an integral over (ox)N. (=X(i) ( N Irt" i=1 f J(ox)N ) ()i(m Y<o (\y 1,htey(Bay1 (ty)) < O dv, (7.17) with t-, restricted to C(T). We will make a change of variables among the units vy that is suited to the structure of the m,, and B-<. (ty). This change will reveal "nesting phenomena" (7.19) among the my which will make the integral over (ox)N easier to evaluate. Changes of Unit Variable for Nesting Coordinates. Recall that m(-y, a) denotes the multiplicity of the simple root a as a summand of y. For a root 7y E Rand another root -y -< yj, call the set of changes of variable for all -y -< -y the changes of unit variable for -y. of these changes by the operator Ny,. We let N>. We denote the application denote the changes for -y' >- -y, starting from -yi. The N signifies that these changes create the nesting coordinates. Now N>.+1 (m.,) is a power of the uniformizer 7r times a product of unit variables. We describe exactly which unit variables appear in this coefficient. Lemma 7.4.1 Let v(y, a) be J v(-y, a) = 0>-y,3#maps to a 68 vo, with an empty product defined to be 1. Then the unit variable coefficient of N",, (m,,j ) is V-Y H v(7Y3+1, a)nt(,j(^),C,). aEDProof The proof is by induction. We note that the lemma is true for all mY,O in which case inYO = t 7 . Fix 0 > j < N and assume it is true for all N yj,(m,j) for - -<_7j+1. Now apply N,, to see that the unit variable coefficient of N>. y,r(my,j) 1 for -y -< 7 3+ is v,[v 7y3 ±v(7Yj+, JJ aj+1)](w(7),a+1) v(Y+ 1, a)m(wJ(7)')) (7.18) aED\aj+1 The j + 1-th step in the Bruhat decomposition is to conjugate uW,( by aY+1 (N".,, (m-,,+)). N>73,j (rmj 3 7 ) (N,+ 1 (m,j)) Using the induction hypothesis on the unit coefficients of ), this conjugation has the effect (in terms of units) of multiplying (7.18) by aj+1)](Wi(7),a+1). [v y+v(7+1, 7 Now rn(wj(-y), %+1) - rn(wj+1(-7), aj+1 ) = (wj(y), aY+i) and v(7yj+, a)m(w(^),a) for all a 54 aj+1. (In the case j = - v(yj+2, a)m(w+l(Y) , a) N - 1, we let J- v(YN+1,a)=1 I3ER-,3mapstoa These facts combined with (7.18) proves the induction step. El 69 Corollary 7.4.2 Let -y E R- map to a E D-. Then the unit variable coefficient of N>,(m,) is 3 maps to Q 13y, Now we may apply N.< 1 to the integrand of (7.17) to get P(N-<(m1171 1 )) = Jl[ (7.19) 'P(v~(ir." + V-1( rn2 + ... + v,-1r"Q-m)...)) aED- Y<O where we have re-indexed the vI, as va,i such that y is the i-th most succeeding root that maps to a, and na,i are integers depending on ii. This is the nesting phenomenon referred to above. The rational functions B,, (-y) are also suited to N-.y<; Lemma ?? describes how. Lemma 7.4.3 Suppose -yj E R- maps to a simple root a. Let 3 be the next root succeeding -yj that also maps to a, if such a root exists. Then for any yy -< , NyjB-O(t,) = v"(w1(7)'")Ba(t,)' where B p(t,)' denotes a function independent of the unit variable vj (if 3 does not exist, then -< 3 is interpreted as E R~). In other words, v , completely factors out of NyB- (ty). Proof For 7 -< /, v',, completely factors out of Nyj(ty) to the power m(wj1((y), a); here we have used that m(wj_(Y), -a) = 0 for any -yj -< -y -< 3. Furthermore, for any path P in R that respects -< with PO -<3 , then v-, also factors out of t(P) to the power m(wj-1(E(P), a). The lemma follows from these facts and the definition of B.<-3 . 0 70 7.5 Final Integration This section proves Theorem 7.1.1 using the main result of section 7.6. The proof motivates a "subvariable" z of a unit variable vY which is treated in that section. For an N-tuple f, consider the integral I(T) given by N X(F) fj 1|7r ni dv,. 11 lht(,) (NR- B-.,y (ty)) O(NR-m,~) /ER( i(.x)N YER ER- v (7.20) Lemma 7.5.1 Let 71 = ht(7y) for some j. (nN, ..- , n 1 ) be an N-tuple of integers. Suppose that nj= Then H ht()(Byj(ty)) IP 7ER- 1pht)(ty) = YER- for t, restricted to C(Ti). Proof The operator B'Y, is a sequence of changes of the form t -+ t, - c(P)t(P) where P is a path in R- with E(P) = -y and Po = -yj. We claim each such change leaves invariant the product H I pht(-y) (tY) . YER~ Indeed, apply the change for one such P to get IIPht(y)(t 7 - c(P)t(P)) H y'ER~,#y 71 Iht(-')(ty,). The conditions on the ty along with the assumption that the valuation of t-, is ht(Y3 ) imply t(P) E pht(Y) as E(P) = y. Since c(P) is an integer, we get mod pht() t1 - c(P)t(P) = t. and the desired invariance is proved. L Definition 7.5.2 Given an N-tuple i, let j = j(W), 1 < j < N, be the greatest index such that for all k < 1, nk = ht(-Yk). Successively applying Lemma 7.5.1 gives JJ tht( 7 )(B y (t-y)) 71 rj 1pt-) (Bayj(ty)) = yER~ YER- The condition 1 ,ht(-)(NR-Byj (tyj)) is simply t£f E Pht(%) so we may assume nj > ht(%y). We claim that (7.20) is 0 unless nj = ht(-yj). Assume nj > ht(-y). Recall the re-labeling of 7j as (a, i), where -yj is the i-th most succeeding root that maps to a. We integrate va,4 over o'. If no root preceding -yj maps to a, then from (7.19) we need to consider V-1 Vir 7rn 72 ; otherwise, we have V- (rn.,j + v-+i1 va~i"" +v,i+lv7r" n) where no,, is the negative of the valuation of M , and v and n are independent of both va,i and va,i+1. We claim naj = ht(yj) - nj - 1 given our condition on j. This may be proved using the form of the m., (Lemma 7.2.3) and induction similar to that in the proof of Lemma 7.4.1. By Lemma 7.4.3, each support conditions 1,ht(-) (NR-B, (ty)) is independent of vcj. Thus the integral of va,i over o' in the first case is 0 by (7.2) because nj > ht(-yj); and likewise in the 1 +v E Pn-ht(fj). In the second case, second case is 0 unless m = nci and 1 + v a,i±1 1 n 3 hly) vc,i+1 must be of the form Vai+1 = -1 + zirnj -ht(-yj) (7.21) for any z E o. This is how the subvariable z of va,i+1 is determined. We would like to integrate z over o to get 0 by (7.3), but the support conditions may depend on z. We assume that the support conditions are independent of z which is proved in Section 7.6. Thus we apply (7.3) to see that the integral is 0 in the second case as well. This proves that l(n) is 0 unless nf = (ht(yN), - ,ht(-y1)). 73 7.6 The Support Conditions Are Independent of z In the notation of section 7.5, we show that the product (7.22) 1 IP t(t)y(NR-B-<7j(t7)) -YER- restricted to C(T), where j = j(f), is independent of the subvariable z of v,,i+1. In the following, a primed function f' will denote the result of applying NR- to f and factoring out the maximal power of v,,i+1 (which may be 0); in particular, primed functions will be independent of v,i+i. Our analysis will depend on the type of the root system R. We first examine the possible levels of paths in R-. Lemma 7.6.1 Let P be a path in R- with P0 =y,. Consider a collection of paths of any level from P such that no two branchingpaths share the same junction. Let m be the number of all junctions. Then m is less than the multiplicity of ac in w 3 (E (P)). Proof Apply wj to the roots in paths in this set to express wj(E(P)) c R- as a sum of other roots in R- with aj appearing at least m + 1 times in the sum, one from each junction and one from P itself. E Type A In type A, simple roots occur with multiplicity at most one. By Lemma 7.6.1, there are only level 0 paths and thus Lemma 7.3.8 gives B = B-,(t) =,(t1) c(P)B<-,j(t(P)) E pht() E r-(P)=-YP0='Y 74 (7.23) Apply NR- to (7.23). The unit variable v,,i+1 factors out of NR- B, power 1, and out of NR-B,,(t(P)) (ty) to the to the power 0 by Lemma 7.4.3. Also for such P, (t(P)) = Bay (t(P)) B, because there are no level 1 paths. Dividing NR- B, S B-<,(ty)' - va-i1 (tx) by v,,i+l gives the condition c(P)B ,y,(t(P))' E pht(y) (7.24) E(P)=-Y,PO=-yj By (7.22), B--<yj(t(P))' E pht')h(7)n Then substituting (7.21) into (7.24) shows that (7.24) is independent of z, using c(P) E Z. Type D The roots in D, r > 4, are of the form ±ej t Ej, where i $ j and where {EI}1<i<;r is a set of r orthonormal vectors. We let R- consist of the of the roots ei t ej where i > j, with simple roots DD- ={ji+1 - Ei1<i<, U E2+E1. The roots in D- that may appear with multiplicity 2 are {Ei+1 - Ei}1<i<,-1 while 75 all other appear with multiplicity at most 1. When aj is a root of the latter type, the analysis done in type A suffices to prove the z independence; when it is of the former, then any path has at most one junction. We analyze the terms arising from such paths and show that, while those terms containing z dependence may not be in Pht( 7 ) individually, the sum of them is. This sum is equation (7.31). Suppose that P is a path in R and Q branches off of P. We write the root sequence for P as {Pprec, EQ, Psucc} where Ppr and Pucc denote the subsequences of P's sequence that precede and succeed EQ, respectively. Not that Psucc is a path in R but Pprec is not necessarily one. We make an abuse of notation and define t(Pprec) to be t(Prec) = t(P) tr(Q)t(PSUCC) As for type A, we now determine how each condition NR-B- I(t-y) E P ht(-y) depends on v, and thus on z. From Lemma 7.3.8, (7.25) B-,,(ty) = B-yj(ty) - Ec(P)By,(t(Pprec))X P B-<yj(tr(Q)) - Apply NR- to (7.25). 1(P -<Q)c(Q)Ba,j(t(Q)) B-<yj(t(Psucc)). Now v,in factors out of NR-By(t.) to the power 2; out of NR-Ba,(tE(Q)) to the power 1; and out of NR-B.<yj(t(Pprec)), NR-B.<y(t(PUcc)), 76 and NR-Bayj(t(Q)) to the power 0. Then v2+NR- B-. yj(ty) = B- y (t7)' v- ,Vic(P)B-. yj(t(Pprec))x - (7.26) P (B.<,(tE(Q) E(P )' - vai+Z Q Now B<73j (tr,(Q))' - va +1 E c(Q)B y,(t(Q))' = vjNR- B-.(ts(Q)). (7.27) Q Therefore we re-write the right side of (7.26) as V3 +1 c(P)B.<y,(t(Pprec))' (v- NR-B.<-yj(tE(Q))) B.<, (t(Psucc))' B<y7 (ty)' P (7.28) + Va+lc(P)B-yj(t(Pprec))'B.y%(t(Psucc))' E 1(Q -< P)c(Q)Bayj(t(Q))' . Q By (7.22), we may assume B.V B-<yj(tF,(Q))) -(pe)'gNR- B <y(t(Psucc))' E Pht(-y)+ht(-yj)-nj using B<^j(t(Pprec)) = B j(t(Prec)), B.<-yjt(Psucc) = B <yt(Psucc). 77 Applying (7.21) to the first v-j in (7.28), this equation simplifies modulo pht(-y) to B-.<y,(ty)' (V~ iNR- B-.,j(tr,(Q)) B-.<^tj(t(Psucc))' (7.29) V -c(P)B-.<yj(t(Pprec))' P 1(Q -< P)c(Q)B.<, (t(Q))' + v-2c(P)B< 7 , (t(Pprec))'B-yj (t(Psucc))' Q Now, using (7.27), we collect the z-dependent terms in (7.29) modulo pht(-t) to obtain v-z7r"n-ht(7j)B<7j ( 13 E(P, Q)c(P)c(Q)t(Prec)t(Q)t(Psucc)) (7.30) (P,Q) respects -< where we have the following definitions. We say that (P,Q) is a branching pair if P and Q are paths in R with Q branching off P, and that (P,Q) respects the convex ordering if both P and Q respect the convex ordering. Define -1 E(T) c(P, Q) : T 1 ) -< T(' :{1: = -< c(P)c(Q) and t(P, Q) = t(Pprec)t(Q)t( Psucc). The sum in (7.30) is then E(T)c(T)t(T). (7.31) T respects -< We show that this sum is 0 by constructing an involution among the terms in the sum, pairing up each term with one that is its negative. This involution is deter- 78 mined roughly as follows. Each term in the sum arises from a branching pair (P,Q). Essentially, we "transfer" a certain root from one path in this pair to the other, thus constructing a branching pair (P, Q). We first describe a result for the structure constants. For simply-laced root systems R with a, 3 E R, we set cc,, = ca,,0;1,1 following notation of [Spr]. As usual, if either a, # or a+3 V R, then we set c0 ,3 = 0. The next lemma states two standard relations among the structure constants. Lemma 7.6.2 Let R be a simply-laced root system and let a,/3, y be linearly independent roots in R. Then (i) cO = -COa (anti-commutativity) (ii) ca,Oc,+O,y + cOy,cO+y,a + c-y,acyc+, = 0 (Jacobi relation) Proof [Spr]. L We apply relation (ii) to structure constants of the branching pair (P,Q). Lemma 7.6.3 Given a branching pair (P,Q), let a = min(Q) # = E(Q) - min(Q) -y = E(PSUCC). Then exactly one of the sums 3 + -y, -y + a is in R, and caO Ca+,,yy = ca,+,-y cc= :3 + - E R - ca,, co,-Y+a : -y + a E R 79 Proof That exactly one of the sums ,+7, -y+a is in R follows from a straightforward check using the characterization of roots in type D. Therefore, exactly one of the three terms in the Jacobi relation is 0, depending on which sum is not in R, leaving the stated equation between two remaining terms. E Operations on branching pairs. We construct a branching pair (P',Q') out of P and Q by re-arranging the roots in the paths; t(P',Q') = t(P,Q) because all the relevant roots remain unchanged, and c(P', Q') is equal to c(P,Q) up to sign. There are four operations we consider. Commute. Suppose that min(PSUCC) $ P0 . If E(Q) + min(Psucc) V R, then set (P,Q)COrnn = ({Pprec, min(Psucc), E(Q), Psucc\ min(Psucc)}, Q). Clearly c(P,Q)comm is equal to c(P,Q). Transfer to the preceding sequence. In the case E(Q) - min(Q)+ E(Pucc) E R, transfer min(Q) to Pprc; set (p, Q)Prec = ({Pprec,min(Q), E(Q) - min(Q),Psucc}, Q\min(Q)). Then it follows from Lemma 7.6.3 that c(P, Q)prec - c(P, Q). Transfer to the succeeding sequence. In the case min(Q)+E(Ppre) E R, transfer min(Q) to Psucc; set (P,Q)succ = ({Pucc, (Q) - min(Q), min(Q), Pucc}, Q\ min(Q)) 80 Again from Lemma 7.6.3, c(P, Q)sUcc = -c(P, Q). Switch. We set (P,Q)switch Pprec, (Psucc) Q} , Psucc) Then c(P,Q)switch = c(P,Q); indeed, we get one -1 by changing cE(Q),(P-c) to cE(Pmc),E(Q) and another -1 from the E factor. The initial input of R is a branching pair (P,Q) that respects the convex ordering. It generates a sequence of intermediate branching pairs that do not respect the ordering until a pair that does respect the ordering is obtained. This pair (P,Q)R is the final output and satisfies c(P,Q) = -c(P, Q)' and t(P,Q) = t(P,Q)R. Root Transfer Algorithm R for Simply-Laced Root Systems Step 1. Given a branching pair (P,Q), la. If min(Q) -< min(Psucc), lai. If EQ - min(Q) + EPu~c E R, then transfer min(Q) to Pprec. With the resulting pair, go to Step 1. laii. If min(Q)+EP E R, then transfer min(Q) to Pc. With the resulting pair, go to to Step 2. 1b. If min(Pucc) -< min(Q), ibi. If EQ + EPsucc - min(Psucc) E R, then apply the commute operation to min(Psucc) and EQ. With the resulting pair, go to Step 1. 81 1bii. If min(Psucc) + EQ E R, then transfer min(Psucc) to Q, i.e., apply the inverse of the transfer to the succeeding sequence operation to min(PSUCC). With the resulting pair, go to to Step 2. Step 2. Given a branching pair (P,Q), 2a. If either (P,Q) or (P,Q)switch respects the convex ordering, then output that pair. 2b. If max(Pprec)+ EPsucc E R, then apply the inverse of the commute operation to the roots max(Ppre) and EQ in P. With the resulting pair, go to Step 2. 2c. If max(Ppre) +EQ E R, then apply the inverse of the transfer to preceding sequence operation to max(Pprec). With the resulting pair, go to Step 2. Lemma 7.6.4 The root transfer algorithm R for simply-laced R is well-defined and is an involution on branchingpairs (P,Q). Furthermore, t(P,Q) = t(P,Q)R. Proof First observe that during the steps of the algorithm no roots are added to or removed from the union of the root sequences of Q, Pprec, and Psu 1 c. Therefore t(P,Q) = t(P,Q)R. We claim each intermediate pair (P,Q) possesses the following properties: (i) Q respects the convex ordering and P does not, but P absent E(Q) does respect the convex ordering, so that max(Ppr,) -< min(Psucc), max(Ppre) -< min(Q). 82 (ii) E(Psucc) -< min(Psucc), E(Q) -< min(Psucc), E(Psucc) -<min(Q), E(Q) -<min(Q). (7.32) Property (ii) also holds for pairs that do respect the convex ordering. We will use the fact that in simply-laced root systems R, for any a,3 E R, the quantity (a, ,v) lies in {0, ±1, ±2} and satisfies the following conditions: (a,fl a+ f E R < a = # <-- -1 (7.33) (af3v) = 2. First we prove that min(Q) -, min(Psucc) assuming that Psucc and Q respect the convex ordering. We may assume that a3 = Ej+1 - Ej for a fixed 0 < i < r - 1, where Po = -yj. Recall that the integer sequences for any path in D, are of the form a = {, ..., 1, 1}, b = {1, ..., 1, ,1, -1}. Then wj_ 1 (E(Q)) and wj_1(E(Pucc)) must be of one of the following forms: {Cm - Ei+i, Ei - Em', Em - Em', Ek + Ekt, -Ei+i + Ei for some m > i + Ii > m', k > i + 1, k > k'. In addition, EQ + EPucc E Dr since P is a path. We directly check the possibilities listed to see that min(Q) cannot equal min(Pucc); thus la and lb are exhaustive. Lemma 7.3.2 applied to the initial input implies that steps lai and 1bi always generate intermediate pairs that possess properties (i) and (ii). Likewise we check that steps laii and 1bii preserve property (ii). 83 We claim that the algorithm must eventually enter step 2. For if it did not, then at some point in the algorithm, we have a pair (P,Q) at step 1 such that either Psuc or Q consists of the single root PO while the other has root sequence {0n, ..., 1, PO} with n > 0 such that /i + Po V R and 31 -< Po,13 +1 -< f3i for all i. By construction, (W, + .. +#1 - PO) + (- PO) E R and thus (On + .. + 01 - PO, -PO)=-. The assumption that the algorithm never enters step 2 implies that (#i, -PO) for each i > 0. =0 Then (PO, PO) = 2 gives a contradiction. This proves that the algorithm must eventually enter step 2. Now, let (P,Q) be an intermediate pair at step 2. If Ppr EQ + EPsuc. is empty, then EP = Therefore either EPsucc -< EP --< EQ or EQ -< EP -< EPsucc by the convex property. We get from these two possibilities that either (P,Q) or (P,Q)switch, respectively, respects the ordering by (ii). If Pprec is not empty, then max(Prec) succeeds at least one of EQ, EPsuc,; oth84 erwise, the convex property would imply the initial pair did not respect the convex ordering. If max(Pprec) succeeds only one of the two roots, then by (ii) either (P,Q) or (P,Q)switch respects the convex ordering. If max(Pprec) succeeds both, then neither (P,Q) nor (P,Q)switch respects the convex ordering and the pair enters step 2b or 2c, which we check are exhaustive. As before, we also see that 2b and 2c preserve (i) and (ii). The algorithm terminates because 2a and 2b decrease the number of roots in Pprec, and we proved that the algorithm terminates when Ppr is empty (though Pprec being empty is not necessary for termination). We must show that (P,Q) # (P,Q)'. The root transferred at step laii or 1bii, call it 'y, ensures that the pair output at step 2a is distinct from the intitial output pair, except possibly in the case that P and Q have identical roots at the places succeeding -y. But in that case, we get that - + 2/3 E R for some root 3, a contradiction in simply-laced R (For non-simply laced R, there is a modification of the algorithm at this point). To check that the algorithm is an involution, see that step 2b and 2c or inverse to step lai and 1bi, and the root transferred at laii or 1bii is the same root transferred there by applying R to (P,Q)". Type E In E 6 , the simple root 64 - E3 may appear with multiplicity three. Therefore for there are two types of rooted trees of paths in R- that begin at 64 - E3: those with two children of the root, and those with one child of the child of the root. In the support conditions, we add and subtract terms as we did in type D and collect the z dependent terms. The powers of z that appear are z and z2. There is a 85 different vertex-relation coefficient each power of z. There is an equivalence relation on the set of rooted trees that respect the convex ordering. We say T1 ~ T 2 if T can be obtained by applying a sequence of root transfer operators Re, where e is an edge of a rooted tree. The sum of these terms weighted by the structure constants and vertex-relation coefficients is 0. In contrast to Type D, where the number of trees in an equivalence class is always 2, this number for Type E can vary, with possibilities including 4, 5, 8, 9, or 16. The strategy is to characterize all possible equivalence classes and show the sum over each class is 0. The equivalence class for a tree with m edges is described by an m-regular graph. Below we list the vetex relation coefficients vr 1 and vr 2 in E 6 . vri(T) = -2 : T(1) 1 : ( 1 :T(1,2) -PT1,1) -< and TM T(1,1) and V(2) T-< () and T(1,1) -<T and < T() :(1) 1 : V(,') -< V() and otherwise 86 -(1,2) and T(1,1) -2 0 (1,2) -< (1,1) T(M <T(1,1,1) T(1,1,1) -1 : T(1 ) - T('1' and TM --< P,2) 1 : TOl) -<T(',) and _- 7l) T(1 ) -< T(1 ,2) and T 1'1 -< TI) 1 1 r2(T) rT1,2) -< T 112' and T 1' 1) -<T) 1) -< T(1 1)and T 1 1 )-<rl) 1,1 0 otherwise Root Transfer Algorithm for Type B The root transfer algorithm for Type B extends that for Type D with some modifications. As in type D, we use the characterization of roots in Br. Recall that the positive roots of B, consists of the vectors ej ± ej and Ej, j > i roots Ei+1 - Ej, i > > 0 with simple 0, and Eo. The coefficient sequences for paths may now include ±2. We describe the points in the type D algorithm that are modified. The first modification deals with inputs (P,Q) for which the algorithm outputs two pairs instead of one. That is, in type D we could partition the sum into sets of two terms such that the two terms add to 0. But in type B, we partition the sum into sets of two or three terms, such that the terms in a triple add to 0 as well. The following conditions are to be placed at the indicated spots in the algorithm for type D. Here, (P 1 , Qi) always denotes a pair where min(Qi) if of the form ei and has coefficient 2 in Qi; (P 2 , Q2) a pair where min(P 2 sc) coefficient 2 in P 2 succ; and (P 3, Q3) a pair where min(P 3 suc) form ej and has coefficient 1 in both paths. 87 of the form ej and has = min(Q 3 ) is of the laiii. Given a branching pair (P, Qi), if min(Q1) -< min(P succ) and min(Qi) has coefficient 2 in Q1, then transfer min(Q1 ) to Pic to make the pair (P 2 , Q2), where Q2 = Q1\ min(Qi) and P 2 'ucc = {min(Q 1 ), P1 suc} but with min(Qi) having coefficient 2 in P 2 succ. In addition, transfer one copy of min(Q1) from Q, to Pi succ to make (P 3 , Q3), where Q3 = Q, but with min(Q1) having coefficient 1, and P 3 = P 2 but with min(Qi) having coefficient 1 also. With these two pairs, go to Step 2. lbiii. Given a branching pair (P2 , Q2), if min(P 2 suc) -< min(Q 2) and min(P 2 succ) has coefficient 2 in P 2 ucc, then transfer min(P 2 SUCC) to Q2 to make the pair (P 1 , Q1), where Q, = {min(P 2 succ), Q2} but with min(P 2 succ) having coefficient 2 in Qi; and where P'succ = P 2 succ\ min(P 2 succ)}. In addition, transfer one copy of min(P2succ) from min(P2succ) to Q2 to make (P 3 , Q3), where Q3 = Q, but with min(P 2 succ) having coefficient 1, and P 3 = P 2 but with min(P 2 succ) having coefficient 1 also. With these two pairs, go to Step 2. 1c. Given a branching pair (P 3 , Q3), if min(P3 succ) = min(Q 3 ) = 7, then -Y must have coefficient 1 in both paths. Construct two pairs (P, Qi) and (P 2 , Q2) out of (P,Q). Here Q, = P1 succ = P 3 succ\7 Q3 but with -y having coefficient 2 in Q1, and Furthermore Q2 = Q 3\7 and P 2 SUM = P3 succ but with y having coefficient 2. With these two pairs, go to step 2. The second modification deals with terms in the sum that do not correspond to pairs of paths, but to paths with a 2 in the b sequence. Specifically, let S be a path with root sequence {..., Sn+1 , Sn., } and coefficient b, = 2 such that ESO,n is of the form Ej and ESo,n is of the form 88 ej + Ej. #,+1 is of the form Esj - ej. Let (P,Q) be a branching pair with Pprec = {..., Sn+3, Sn+2} Q = {Sn+1, Sn, ---} Psucc = {Sn, Sn- 1 , ... } 2c(s) = c(P,Q). Apply Step 2 of the algorithm for type D to (P,Q). The output corresponds to the term that cancels the term that corresponds to S. The third modification deals with a path S such that So = E0, b1 = -2, and S1 = Ej + 6o. Let (P,Q) be the branching pair Pprec = {..., S3,32} Q = {S1, So} Psucc = {So} Apply Step 2 of the algorithm for type D to (P,Q). The output corresponds to the term that cancels the term that corresponds to S. Type C The root transfer algorithm for Type C also extends that for Type D with some modifications. We use the characterization of roots in C,. Recall that the positive roots of C, consists of the vectors ej t ei j > i > 0 with simple roots Ei+1 - ei, i > 0, and 2eo. The coefficient sequences for paths may now include +2. We describe the points in the type D algorithm that are modified. The first modification, as in Type B, deals with pairs (P,Q) such that min(Q) = min(Psucc) = 7. In this case, -y is of the form ej - Ei, j > Z. 1c. Given a branching pair (P, Q), if min(Q) = min(PSUCC) = 7, then construct the 89 pair (P',Q') such that Pprec = {Pprec, } Q'I= QVY PS'ucc = PSUCC\-Y with -y having coefficient 2 in P'. With this pair, go to Step 1. 2b. Given a branching pair (P,Q), if max(Pprec has coefficient 2 in P, then construct the pair (P',Q') such that Pprec = Pprec\7 Q'={py,Q} PSUCe = {Y, Psucc} with y having coefficient 1 in Q and P. The second modification deals with pairs (P,Q) such that min(min(Q), min(Pucc)) Ej- and EQ + EPsucc = ej + Ej. In this case, both conditions of lai and laii (or 1bi and 1bii) may hold; therefore go to the following laii (or lbiii). laiii. Given a branching pair (P, Qi), if EQ1-min(Qi)+EPisuc E R and min(Qi)+ EPi ,u, E R, then construct the following pairs (P 2 , Q2) and (P 3 , Q3). Here, 90 = P2 prec = Q2 - P1 prec Q min(Q1) P 2 succ = {min(Q), Pi succ} P3 prec {min(Q), P and = prec} 3 =Qi\ min(Q1) P3 succ P1 succ- With (P 2 , Q2) go to Step 2, and with (P 3 , Q3) go to Step 1. 1biii. Given a branching pair (P 2 , Q 2 ), if EQ 2 + EP 2 SUCC - min(P 2 succ) E R and min(P2 succ)) + EQ 2 E R, then construct the following pairs (P, Qi) and (P 3 , Q 3). Here, PI prec = P2 prec Q1 = {min(P 2 succ), Q 2} P succ = P 2 succ\ min(P2 succ) 91 and P3 prec = {P2 prec, min(P2succ)} Q3=Q2 P3succ = P2succ\ min(P2succ). With (P1 , Qi), go to Step 2. With (P 3 , Q3) go to Step 1. 2c. Given a branching pair (P 3 , Q3), if max(P 3 prec) + EQ 3 E R and max(P 3 prec) + EP3 suc E R, then construct two pairs (P, Qi) and (P 2 , Q2) such that P 1 prec Q1 P1 succ - P3 prec\ max(P 3 prec) = {max(P3 prec), Q3} P3 succ and P2 prec - P3 prec\ max(P3 prec) Q2 = Q3 P2succ = {max(P3 prec), P3succ}. With (P 1 , Q1 ) and (P 2 , Q2 ), go to Step 2. The third modification deals with terms that correspond to a single path instead of a pair. Let S be a path such that ESo,n = Ej - ej and EO,n+1 = 2e. 92 Construct the pair (P,Q) such that Pprec = {---Sn+3, Sn+2} Q = {Sn+1 ,Sn} Psucc = {Sn, Sn- 1 , ... } Apply Step 2 of the algorithm to (P,Q), The output then corresponds to the term in the sum that cancels the term corresponding to S. 93 94 Chapter 8 The Matrix Coefficient S(p(g) - q1) We present some results about the matrix coefficient S(p(g) - #gO). This coefficient on the (U, 4) side is analogous to W(p(g) - OK) on the K-fixed side in that it shares a similar evaluation at torus elements. A full treatment should appear in a separate document. Recall that W(p(a) - #,5 .i) is a monomial in C(Xv); this was the main result of chapter 7. We next show that S(p(a) - U;1) and S(p(a) -OK;1) are also a monomials. We compare the calculation of S(p(a) - #K;1) to its analogue W(p(a) - oUp;1 ' Theorem 8.0.5 Let a = aoqrX with ao E T(o) and A J E X' dominant. Then 2 uz;1 (ka) dk = (1 - q-1)d%1/ X(a) 11 T E R+ K -q1i-7r. V and K OK;1(ka) dk = (1 - where d is the dimension of X'. 95 2 q1)djB 1/ -(a)q-|R+I Proof The supports of #g .1 and qK;1 are BIht and BI, respectively; in addition, BIht C BI. We thus determine when kwrA E BI. We have, from the Iwahori decomposition of K, K = B(o)n,(w- 1 (U-) n U- n I)(w- 1 (U-) n U n I). The dominance of A implies 7r-A (w-1(U-) n U n I)r-' E I. Therefore, after conjugating by ArA, we are left to determine when (nw(w-1 (U~) n U-)) n BI is non-empty. Lemma 8.0.6 says that this set is non-empty only if w = 1. integral for S(p(a) - #u, .1) now follows from the result for W(p(a) - # The 1) of chapter 7. The result for S(p(a) - OK;I), however, now follows simply because the measure of BI is (1 - q-1)dq-lR+I. In this case of OK,1, there is only one non-zero value of x as k ranges over K; contrast to the case of # ,0;1, where x takes on infinitely many values as k ranges over K, just as when u ranges over U- in chapter 7. 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