Calculating Canonical Distinguished Involutions
in the Affine Weyl Groups
Tanya Chmutova and Viktor Ostrik
CONTENTS
1. Introduction
2. Background
3. Theorems
4. Tables
References
Distinguished involutions in the affine Weyl groups, defined by
G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called
canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine
Weyl group contains precisely one canonical distinguished involution. We calculate the canonical distinguished involutions
in the affine Weyl groups of rank ≤ 7. We also prove some
partial results relating canonical distinguished involutions and
Dynkin’s diagrams of the nilpotent orbits in the Langlands dual
group.
1. INTRODUCTION
2000 AMS Subject Classification: Primary 17B20; Secondary 20H15
Keywords: affine Weyl groups, cells, nilpotent orbits in semisimple
Lie algebras
It is well known that the Kazhdan-Lusztig combinatorics
of the affine Hecke algebra is related to the geometry
of the corresponding algebraic group (over the complex
numbers) G in a fundamental way. The central result
is the deep theorem of George Lusztig establishing a bijection between the set U of unipotent classes in G and
the set of two-sided cells in the corresponding affine Weyl
group Wa , see [Lusztig 89]. Using this result, one defines
a map from U to the set X+ of the dominant weights in
the following way: let O be an unipotent orbit and let cO
be the corresponding two-sided cell. George Lusztig and
Nanhua Xi attached to cO a canonical left cell CO , see
[Lusztig 88]. In turn, the left cell CO contains a unique
distinguished involution dO ∈ Wa , see [Lusztig 87], which
is the shortest element in its double coset W dO W with respect to the finite Weyl group W ⊂ Wa . It is well known
that the set of double cosets W \Wa /W is bijective to the
set X+ since any such coset contains a unique translation
by a dominant weight. Combining the maps above, we
get a canonical map L : U → X+ . The explicit calculation of this map is equivalent to the determination of
the distinguished involutions lying in the canonical cells
(or, equivalently, those cells which are shortest in their
c A K Peters, Ltd.
°
1058-6458/2001 $ 0.50 per page
Experimental Mathematics 11:1, page 99
100
Experimental Mathematics, Vol. 11 (2002), No. 1
left coset with respect to the finite Weyl group). We call
such involutions canonical distinguished involutions. We
believe that understanding these involutions is an important step towards the understanding of all distinguished
involutions and cells in the affine Weyl group.
In [Ostrik 00], one of the authors suggested an algorithm for the calculation of L, and now this algorithm
is known to be correct thanks to the (unfortunately still
undocumented) work of R. Bezrukavnikov. The aim of
this paper is to present results of calculations made using
this algorithm.
In Section 2 we present the necessary background. In
Section 3, we present our main results: the calculation
of the map L for the group GLn and partial results for
other groups. These results are apparently known to the
experts but, to the best of our knowledge, they have never
been published. In Section 4, we present tables giving the
results of explicit calculation of the map L for groups of
small rank. These tables should be considered as a main
result of this work.
2. BACKGROUND
2.1 Notations
Let G be a semisimple algebraic group over the complex
numbers. Let g denote the Lie algebra of G and let N ⊂ g
denote the nilpotent cone. As a G−variety, N is isomorphic to the subvariety of unipotent elements of G via the
exponential map but it has one additional property–an
obvious action of C∗ by dilations commuting with the
G−action. We will consider N a G × C∗ −variety via the
following action: (g, z)n = z −2 Ad(g)n for (g, z) ∈ G× C∗
and n ∈ N .
The variety N consists of finitely many G−orbits,
[Collingwood, McGovern 93]. These orbits, called nilpotent orbits, are the main subject of our study. Any
nilpotent orbit O is identified via its Dynkin diagram defined as follows: let e ∈ O be a representative. By the
Jacobson—Morozov Theorem, it can be included in the
sl2 −triple (e, f, h) (i.e., [h, e] = 2e, [h, f ] = −2f, [e, f ] =
h). The semisimple element h is uniquely defined up
to G−conjugacy by O and vice versa, [Collingwood, McGovern 93]. Let h ⊂ g be a Cartan subalgebra, let R ⊂ h∗
be the root system, then choose a subset R+ ⊂ R of positive roots and let {αi , i ∈ I} be the set of simple roots (I
is the set of vertices of the Dynkin diagram of g). The element h is conjugate to a unique h0 ∈ h such that αi (h0 )
is positive for any i ∈ I and, moreover, αi (h0 ) ∈ {0, 1, 2},
see [Collingwood, McGovern 93]. Thus any nilpotent orbit can be identified by labeling the Dynkin diagram of
g by numbers 0, 1, 2. This is called the (labeled) Dynkin
diagram of g. We note that the Dynkin diagram h0 is
naturally an integral dominant coweight for group G.
Let X denote the weight lattice of G and let X+ denote
the set of dominant weights.
2.2 Equivariant K-Theory of the Nilpotent Cone
In this section, we review [Ostrik 00]. Let KG×C∗ (N )
denote the Grothendieck group of the category of G ×
C∗ −equivariant coherent sheaves on N . It has an obvious structure as a module over the representation ring
Rep(C∗ ) of C∗ . Let v denote the tautological representation of C∗ . Then Rep(C∗ ) = Z[v, v −1 ] and the rule
v 7→ v −1 defines an involution :̄Rep(C∗ ) → Rep(C∗ ).
In [Ostrik 00] the following were constructed:
1. the basis {AJ(λ)} of KG×C∗ (N ) over Rep(C∗ ) =
Z[v, v −1 ] labeled by dominant weights λ ∈ X+ .
2. a Rep(C∗ )−antilinear involution KG×C∗ (N ) →
KG×C∗ (N ), x 7→ x̄.
Then, using usual Kazhdan-Lusztig machinery, the
basis {C(λ)} was defined. Thus C(λ) is a unique selfP
dual element of the form AJ(λ)+ µ<λ bµ,λ AJ(µ) where
bµ,λ ∈ v −1 Z[v −1 ]). The main conjecture is that for any
λ ∈ X+ the support of C(λ) is the closure of a nilpotent
orbit Oλ and that C(λ)|Oλ represents, up to sign, the
class of an irreducible G−equivariant bundle on Oλ . In
this way, we can recover Lusztig’s bijection between dominant weights X+ and pairs consisting of a nilpotent orbit
and G−equivariant irreducible bundle on it (see [Lusztig
89], [Bezrukavnikov 00], [Bezrukavnikov 01], [Vogan 00]
for various approaches to Lusztig’s bijection). All these
conjectures are now known to be true thanks to the work
of R. Bezrukavnikov.
Now let eλ ∈ KG (N ) be the image of AJ(λ) under the
forgetting map KG×C∗ (N ) → KG (N ). By definition, eλ
can be constructed as follows: let L(λ) be the line bundle on G/B corresponding to the weight λ (we choose
the notation in such a way that H 0 (G/B, L(λ)) 6= 0 for
dominant λ); then eλ = [sp∗ π ∗ L(λ)] where π : T ∗ G/B →
G/B is the natural projection and sp : T ∗ G/B → N is
the Springer resolution. (Here sp∗ denotes the direct image in the derived category of coherent sheaves or, equivalently, [sp∗ π ∗ L(λ)] is the alternating sum of the higher
direct images Ri sp∗ .) This definition makes sense for any
(not necessarily dominant) weight λ and we know that
ewλ = eλ for any w ∈ W, λ ∈ X.
Chmutova, Ostrik: Calculating Canonical Distinguished Involutions in the Affine Weyl Groups
101
2.3 McGovern’s Formula
3.1 Richardson Resolutions
We are especially interested in weights corresponding to
the trivial bundles on nilpotent orbits under Lusztig’s
bijection. One of the conjectures in [Ostrik 00] states
that these weights are exactly L(O) with L as defined in
Section 1. Moreover, C(L(O)) represents a class j∗ (CO )
where j : O → N is the natural inclusion and CO is the
trivial G−equivariant bundle on O. Again, this is known
to be true thanks to the work of R. Bezrukavnikov.
There is a simple formula for j∗ (CO ) due to W. McGovern: Let h be the Dynkin diagram of the orbit O.
L
Then h defines a grading of Lie algebra g =
i∈Z gi
where gi = {x ∈ g|[h, x] = ix}. Furthermore, g≥0 =
L
L
i≥0 gi is a parabolic subalgebra of g and g≥2 =
i≥2 gi
is a module over g≥0 . Let G≥0 be a parabolic subgroup with Lie algebra g≥0 and let M = G ×G≥0 g≥2
be the homogeneous bundle on G/G≥0 corresponding
to the G0 −module g≥2 . There is a natural map r :
G ×G≥0 g≥2 → g. The image of r is exactly Ō and,
moreover, this map is proper and generically one to one
and therefore it is a resolution of singularities of Ō [McGovern 89]. McGovern proved that [j∗ CO ] = [r∗ CM ].
Now let R+,0 (resp. R+,1 ) be the subset of all positive
roots such that α(h) = 0 (resp. α(h) = 1). Then the
Koszul complex gives the following formula:
Y
(e0 − eα )
(∗)
[j∗ CO ] =
It follows from the theorem of Hinich and Paniushev
[Hinich 91], [Panyushev 91] on the rationality of singularities of normalizations of closures of nilpotent orbits
that j∗ CO = r∗ CM where r : M → Ō is a resolution of
singularities of Ō. One obtains McGovern’s formula from
this using the canonical resolution of a nilpotent orbit.
Now, let P be a parabolic subgroup of G. The image of
the moment map mP : T ∗ G/P → g∗ = g is the closure
of a nilpotent orbit O(P ) and by a well known theorem
due to R. W. Richardson the map mP : T ∗ G/P → O(P )
is generically finite to one.
Let G∨ be a Langlands dual group of G. By definition
there is a bijection between simple roots of G and G∨ . In
∨
particular, we can attach a Levi subgroup L∨
P ⊂ G to
any parabolic subgroup P ⊂ G. Recall that any coweight
for G∨ is by definition a weight for G.
α∈R+,0 ∪R+,1
(see [McGovern 89] for details). This suggests the following algorithm for computing L(O):
1. Multiply the brackets on the right hand side of (*)
using the usual rule, eλ eµ = eλ+µ .
2. In the expression derived from Step 1, replace each
eλ by ewλ where w ∈ W and wλ is dominant. Then,
make all possible cancellations. (Warning: Step 1
and Step 2 do not commute!)
3. In the expression from Step 2, find the leading term
±eλ such that for any other term eµ the inequality
λ > µ holds (the existence of such λ is a consequence
of the results in [Bezrukavnikov 01]). This λ is exactly L(O).
Unfortunately this algorithm is completely impractical
for groups of large rank.
3. THEOREMS
The results of this section are probably well known to
experts. Moreover, Theorem 3.5 was stated in [Ostrik
00] without proof.
Theorem 3.1. Let P ⊂ G be a parabolic subgroup such
that the map mP : T ∗ G/P → O(P ) is birational. Then
L(OP ) is equal to the Dynkin diagram of the principal
nilpotent element in L∨
P.
Proof: From the above remarks, we know that j∗ COP =
m∗ CT ∗ G/P . Let R+ (L) ⊂ R+ denote the subset of positive roots of the subgroup L. Again the Koszul complex
gives a formula
Y
(1 − eα )
m∗ CT ∗ G/P =
α∈R+ (L)
interpreted in the same way as McGovern’s formula (SecP
tion 2.3). Let ρL = 12 α∈R+ (L) α and let WL ⊂ W
denote the Weyl group of L. The Weyl denominator formula gives
X
Y
(1 − eα ) =
det(w)eρL −wρL .
α∈R+ (L)
w∈WL
In particular we have a leading term e2ρL corresponding
to the summand with w = w0 (L)–the element of largest
length in WL . It cannot cancel with anything else since
the scalar product h2ρL , 2ρL i is clearly bigger than any
scalar product hρL − wρL , ρL − wρL i for w 6= w0 (L).
For the same reason, this term is the unique candidate
for the leading term since λ > µ for dominant λ and
µ which implies hλ, λi > hµ, µi. Since we know that
the leading term exists (this is Bezrukavnikov’s result as
we mentioned in Section 2.3) it should be equal to exρL
where x ∈ W is such that xρL is dominant.
Finally weight 2ρL , considered as a coweight for G∨ ,
is clearly the Dynkin diagram of the regular nilpotent
102
Experimental Mathematics, Vol. 11 (2002), No. 1
element in L∨ since the Dynkin diagram of a regular
nilpotent element is always the sum of positive coweights,
see [Kostant 63].
3.2 Nilpotent Orbits
We will say that a nilpotent orbit is a strongly Richardson
orbit if it admits a desingularization via the momentum
map mP : T ∗ G/P → Ō. We have the following
Theorem 3.2. Let O = OP be a strongly Richardson orbit. Suppose that the G∨ −orbit O 0 of a regular nilpotent
∨
element of L∨
P is also strongly Richardson in G . Then
0
L(O ) = Dynkin diagram of O and L(O) = Dynkin diagram of O0 .
Proof: The equality L(O) = Dynkin diagram of O 0 follows immediately from Theorem 3.1.
Both nilpotent orbits O and O0 being Richardson orbits are special, see [Spaltenstein 82], [Collingwood, McGovern 93]. N. Spaltenstein defined an order reversing
involutive bijection d between the sets of special nilpotent
orbits for G and G∨ . It follows from the definition that
d(O 0 ) = O [Spaltenstein 82]. Consequently d(O) = O0
and equality L(O0 ) = Dynkin diagram of O follows by
symmetry.
Note that not any Richardson orbit is strongly
Richardson. For example the Richardson orbit for Sp2n
attached to the parabolic with Levi factor GL1 × Sp2n−2
is not strongly Richardson for n > 2 (we are grateful to
the referee for showing us this example).
A nilpotent orbit is called even, if its Dynkin diagram
is divisible by 2 as an element of the coweight lattice,
see e.g., [Collingwood, McGovern 93].
Corollary 3.3. Suppose the orbit O is even and that L(O)
is divisible by 2 as an element of the weight lattice. Then
L(O) is the Dynkin diagram of some nilpotent orbit O0
for G∨ and L(O 0 ) = Dynkin diagram of O.
Proof: This is clear since even nilpotent orbits are
strongly Richardson, see [McGovern 89].
A lot of examples of this corollary can be found in the
tables at the end of this paper. In fact, this is the only
general statement we can prove (or even just formulate)
about these tables.
It is well known that nilpotent orbits in GLn are numbered by partitions of n: to the partition p1 ≥ p2 ≥ . . .
one associates an orbit of nilpotent matrices with Jordan
blocks of size p1 , p2 , . . .. One finds the Dynkin diagram
of the orbit O = O(p1 , p2 , . . .) as follows [Collingwood,
McGovern 93]: order n numbers p1 − 1, p1 − 3, . . . , −p1 +
1, p2 − 1, . . . , −p2 + 1, . . . in decreasing order, then the
label of i−th vertex of the Dynkin diagram will be the
difference of the i−th and i + 1−th numbers.
The following lemma is well known.
Lemma 3.4. For G = GLn and a parabolic subgroup
P ⊂ G, the map mP : T ∗ G/P → O(P ) is birational.
Proof: The isotropy group of any nilpotent element in
GLn is connected.
Let O = O(p1 , p2 , . . .) be a nilpotent orbit. Let
p01 , p02 , . . . be a partition dual to p1 , p2 , . . . and let O 0 =
O(p01 , p02 , . . .) be the orbit dual to O.
Theorem 3.5. The weight L(O) is equal to the Dynkin
diagram of the dual orbit O 0 considered as a weight for
GLn .
Proof: It is well known that any nilpotent orbit O in G =
GLn is a Richardson orbit and it follows from Lemma
3.4 that it is strongly Richardson. It follows from the
proof of Theorem 3.2 that L(O) is the Dynkin diagram of
d(O) (here d is the Spaltenstein duality). Now the result
follows from the description of d for the group GLn in
[Spaltenstein 82].
3.3 Remarks
(i) As we mentioned in Section 1, the calculation of the
map L is equivalent to the calculation of canonical
distinguished involutions in the affine Weyl group.
For type An , canonical distinguished involutions
were recently calculated by Nanhua Xi [Xi, 00] by
a completely different method. While his result coincides with ours, he does not mention a relation
with Dynkin diagrams of nilpotent orbits.
(ii) For a group G of type different from An , it is not true
in general that the weight attached to a nilpotent orbit and considered as a coweight for the Langlands
dual group G∨ is a Dynkin diagram for some nilpotent orbit of G∨ (see the tables at the end of this
paper). Calculations made by Pramod Achar (private communication) suggest some evidence for the
positive answer to the following question.
Question. Is it true that any Dynkin diagram considered as a weight for the dual group corresponds under
Chmutova, Ostrik: Calculating Canonical Distinguished Involutions in the Affine Weyl Groups
Lusztig’s bijection to a local system on some nilpotent
orbit? (Here a local system is a coherent equivariant
sheaf on the nilpotent orbit such that the corresponding
representation of the isotropy group factors through a
finite quotient.)
so9
Partition
Diagram
(9)
qqq
> q qqq
>q 0
2
(7, 12 )
(5, 3, 1)
The tables contain results of our calculations of the map
L for groups of small rank. We have almost complete
results for groups of rank ≤ 7 (there are some gaps for
groups of types B6 , C6 , B7 , C7 and E7 ) and partial results in rank 8 (for groups D8 and E8 ). The tables are
organized as follows: the tables for classical groups consist of 4 columns: in the first column we give the partition identifying the nilpotent orbit O (see e.g., [Collingwood, McGovern 93]); in the second column we give the
Dynkin diagram of O; in the third column we give the
weight L(O), and the final column contains the square of
the length of L(O) (we normalize the scalar product by
(α, α) = 2 for a short root α). For exceptional groups,
the tables consist of 3 columns: the first column contains
the Dynkin diagram of the nilpotent orbit O; the second
column contains L(O), and the third column contains
the square of the length of L(O). In a few cases, we were
unable to calculate L(O) but we could predict the value
L(O) using duality reasoning. Such cases are marked by
a question mark.
(42 , 1)
0
(5, 22 )
2
0
(5, 14 )
q
>q
2
(3, 12 )
2
2
(32 , 13 )
2
(22 , 1)
0
0
0
(15 )
1
0
0
0
0
2
0
(7)
Diagram
qq
>q
2
(5, 12 )
2
(32 , 1)
2
0
0
1
0
0
qq
>q
0
(17 )
0
qq
>q
2
(22 , 13 )
2
qq
>q
1
(3, 14 )
2
qq
>q
0
(3, 22 )
2
qq
>q
Partition
(11)
(7, 3, 1)
(5, 32 )
q
> q 20
(7, 14 )
0
1
0
qq
>q
0
0
0
0
0
(5, 22 , 12 )
qq
>q 6
2
(32 , 22 , 1)
qq
> q16
0
(5, 16 )
qq
> q20
0
(32 , 15 )
qq
> q28
(3, 24 )
2
1
2
1
1
2
qq
> q70
2
2
2
1
0
2
0
1
1
0
0
2
1
0
1
0
2
0
0
0
2
0
0
1
1
0
2
0
0
2
1
0
0
2
0
0
2
0
0
2
1
0
1
0
2
0
2
0
0
1
1
1
1
2
0
0
0
2
2
1
0
1
0
0
2
1
1
2
0
0
0
2
2
2
2
Weight
2
2
2
2
0
0
0
0
0
2
2
2
0
1
0
0
0
0
2
0
2
0
0
0
1
0
0
2
0
2
0
0
0
0
0
2
2
1
0
1
0
2
0
0
0
qqqq
> q qqqq
> q18
0
0
2
0
0
1
1
0
0
qqqq
> q qqqq
> q20
2
2
0
0
2
1
0
0
0
qqqq
> q qqqq
> q24
0
2
0
0
2
0
0
1
0
qqqq
> q qqqq
> q26
1
1
0
1
0
0
1
1
0
qqqq
> q qqqq
> q32
0
qq
>q 2
0
0
qqqq
> q qqqq
> q16
0
(42 , 13 )
(33 , 12 )
0
0
qqqq
> q qqqq
> q10
2
qq
>q 0
0
0
qqqq
> q qqqq
>q6
2
2
0
2
qqqq
> q qqqq
>q2
2
1
0
qqqq
> q qqqq
>q0
2
2
0
0
Diagram
0
(42 , 3)
0
0
so11
q
> q 10
Weight
0
qqq
> q qqq
> q168
0
(5, 3, 13 )
Partition
1
qqq
> q qqq
> q 78
(19 )
(7, 22 )
so7
0
qqq
> q qqq
> q 70
(22 , 15 )
0
2
2
qqq
> q qqq
> q 60
(3, 16 )
q
>q 2
1
q
>q
2
qqq
> q qqq
> q 40
(24 , 1)
(52 , 1)
1
q
>q
0
qqq
> q qqq
> q 24
(3, 22 , 12 )
q
>q 0
0
q
>q
0
qqq
> q qqq
> q 20
2
(5)
0
qqq
> q qqq
> q 18
2
Weight
0
qqq
> q qqq
> q 16
(33 )
(9, 12 )
so5
2
qqq
> q qqq
> q 14
2
Diagram
2
qqq
> q qqq
>q 6
2
Partition
2
qqq
> q qqq
>q 2
2
4. TABLES
Weight
2
0
1
0
1
1
0
0
2
qqqq
> q qqqq
> q36
0
0
2
0
0
1
0
1
1
0
qqqq
> q qqqq
> q40
2
1
0
1
0
0
2
0
1
0
qqqq
> q qqqq
> q60
0
1
0
1
0
1
1
1
1
0
qqqq
> q qqqq
> q70
2
2
0
0
0
2
2
1
0
0
qqqq
> q qqqq
> q74
0
2
0
0
0
2
2
0
0
2
qqqq
> q qqqq
> q80
1
0
0
0
1
0
2
0
2
0
103
104
Experimental Mathematics, Vol. 11 (2002), No. 1
so11
Diagram
Partition
2
4
(3, 2 , 1 )
(33 , 14 )
qqqq
> q qqqq
> q168
(5, 22 , 14 )
qqqq
> q qqqq
> q176
(32 , 22 , 13 )
qqqq
> q qqqq
> q330
(3, 24 , 12 )
1
0
1
2
0
2
0
1
2
(13)
(11, 12 )
qqqqq
> q qqqqq
>q6
22 , 19 )
qqqqq
> q qqqqq
> q10
(113 )
2
(5, 7, 1)
(5 , 3)
(4, 2 , 1)
0
0
0
0
1
2
2
1
0
1
0
2
0
0
0
2
0
0
2
0
0
1
1
0
0
2
0
2
0
1
1
0
0
0
0
2
2
2
0
0
2
1
0
0
0
2
0
0
2
0
0
1
0
0
1
2
0
2
0
0
2
0
0
1
0
2
0
2
0
0
2
0
0
0
0
0
1
1
0
1
0
0
0
2
0
0
0
2
0
0
1
0
1
1
0
2
1
0
1
0
0
2
0
1
0
1
1
0
1
0
1
0
0
1
1
0
1
0
1
0
1
1
1
1
0
0
0
2
0
0
0
0
2
0
0
2
0
0
0
1
1
1
1
0
0
2
2
0
0
0
2
2
1
0
0
0
2
0
0
0
2
2
0
0
1
1
0
0
0
1
0
2
0
2
0
2
1
0
1
0
0
0
2
0
2
0
0
0
2
0
1
0
0
2
1
1
0
0
1
0
0
0
0
2
1
0
1
1
0
1
0
0
2
0
2
0
1
0
1
0
0
1
1
1
1
1
0
0
1
0
0
2
0
2
1
0
0
0
0
2
2
2
1
0
0
0
0
0
2
2
2
0
0
2
0
0
1
1
1
1
1
1
2
0
0
0
0
0
2
2
2
2
1
0
qqqqq
>q
1
0
1
0
0
0
qqqqq
>q
0
0
0
1
0
0
qqqqq
>q
0
1
0
0
0
0
qqqqq
> q qqqqq
> q572
0
0
0
0
0
0
2
2
2
2
2
2
so15
0
0
2
0
0
0
2
0
0
0
0
0
2
2
0
0
0
qqqqq
> q qqqqq
> q82
0
1
0
qqqqq
> q qqqqq
> q80
2
(42 , 15 )
0
qqqqq
> q qqqqq
> q74
2
(5, 24 )
2
qqqqq
> q qqqqq
> q70
2
(5, 3, 15 )
0
qqqqq
> q qqqqq
> q64
0
(7, 16 )
2
qqqqq
> q qqqqq
> q60
0
2
0
qqqqq
> q qqqqq
> q60
2
(34 , 1)
0
qqqqq
> q qqqqq
> q44
0
(5, 3, 22 , 1)
1
0
qqqqq
> q qqqqq
> q40
2
(42 , 3, 12 )
0
0
0
qqqqq
> q qqqqq
> q36
2
(7, 22 , 1)
0
0
0
0
qqqqq
> q qqqqq
> q32
1
(5, 32 , 12 )
0
1
0
0
qqqqq
> q qqqqq
> q24
0
(5, 42 )
0
0
0
0
qqqqq
> q qqqqq
> q24
2
(52 , 13 )
0
0
0
0
qqqqq
> q qqqqq
> q22
0
(7, 3, 13 )
2
0
1
0
qqqqq
> q qqqqq
> q20
2
2
0
2
0
0
qqqqq
> q qqqqq
> q18
0
(9, 14 )
2
0
2
2
qqqqq
> q qqqqq
> q18
2
(62 , 1)
0
2
2
2
0
qqqqq
> q qqqqq
> q16
2
(7, 32 )
2
2
2
qqqqq
> q qqqqq
> q10
2
(9, 22 )
2
2
0
qqqqq
> q qqqqq
> q330
2
(24 , 15 )
2
(7, 5, 1)
0
qqqqq
> q qqqqq
>q2
2
2
qqqqq
> q qqqqq
> q182
(3, 110 )
(3, 22 , 16 )
2
(9, 3, 1)
0
qqqqq
> q qqqqq
>q0
2
0
qqqqq
> q qqqqq
> q172
(26 , 1)
Weight
0
qqqqq
> q qqqqq
> q168
(32 , 17 )
Diagram
1
qqqqq
> q qqqqq
> q138
(5, 18 )
Partition
0
qqqqq
> q qqqqq
> q110
2
so13
1
qqqqq
> q qqqqq
> q 90
2
2
0
qqqqq
> q qqqqq
> q 86
0
1
2
0
2
1
Weight
qqqqq
> q qqqqq
> q 82
2
1
2
2
2
0
1
2
2
0
2
1
2
0
0
0
1
0
0
0
2
0
0
0
0
0
1
0
1
0
0
0
0
0
(111 )
2
qqqq
> q qqqq
> q110
2
(22 , 17 )
3
(continued)
Diagram
Partition
(3 , 2 )
0
0
(3, 18 )
Weight
qqqq
> q qqqq
> q 90
1
(24 , 13 )
so13
(continued)
2
Partition
(15)
Diagram
qqqqqq
>q qqqqqq
>q 0
2
2
(13, 1 )
2
2
2
2
2
0
1
0
0
0
0
0
0
2
2
2
0
2
0
0
0
1
0
0
0
0
2
0
2
0
2
0
0
0
0
0
1
0
0
2
0
2
0
2
0
0
0
0
0
0
0
2
2
2
2
1
0
1
0
2
0
0
0
0
0
2
2
0
0
2
0
0
1
1
0
0
0
0
2
2
2
2
0
0
2
1
0
0
0
0
0
0
2
0
0
2
0
0
1
0
0
1
0
0
0
2
0
2
0
0
2
0
0
0
0
1
0
2
0
1
1
0
1
0
0
1
0
0
1
0
1
0
1
1
0
1
0
0
0
2
0
0
0
0
2
0
0
2
0
0
0
0
1
1
0
0
2
0
0
2
0
0
1
0
1
1
0
0
0
qqqqqq
>q qqqqqq
> q36
0
(9, 22 , 12 )
0
qqqqqq
>q qqqqqq
> q36
2
(62 , 13 )
0
qqqqqq
>q qqqqqq
> q34
0
(7, 32 , 12 )
0
qqqqqq
>q qqqqqq
> q32
2
(53 )
0
qqqqqq
>q qqqqqq
> q30
0
(7, 42 )
0
qqqqqq
>q qqqqqq
> q28
2
(62 , 3)
0
qqqqqq
>q qqqqqq
> q22
2
(7, 5, 13 )
0
qqqqqq
>q qqqqqq
> q20
2
(7, 5, 3)
2
qqqqqq
>q qqqqqq
> q18
2
(11, 14 )
2
qqqqqq
>q qqqqqq
> q16
2
(9, 32 )
2
qqqqqq
>q qqqqqq
> q14
0
(11, 22 )
2
qqqqqq
>q qqqqqq
> q10
2
(72 , 1)
2
qqqqqq
>q qqqqqq
>q 6
2
(9, 5, 1)
2
qqqqqq
>q qqqqqq
>q 2
2
(11, 3, 1)
Weight
2
0
2
0
1
0
1
1
0
0
0
0
2
qqqqqq
>q qqqqqq
> q40
2
2
2
1
0
1
0
0
2
0
1
0
0
0
Chmutova, Ostrik: Calculating Canonical Distinguished Involutions in the Affine Weyl Groups
so15
Diagram
Partition
2
2
(5 , 3, 1 )
(26 , 13 )
qqqqqq
>q qqqqqq
> q 60
(24 , 17 )
qqqqqq
>q qqqqqq
> q 60
(22 , 111 )
qqqqqq
>q qqqqqq
> q 64
(115 )
2
(52 , 22 , 1)
0
(42 , 32 , 1)
(7, 3, 15 )
(52 , 15 )
4
(7, 2 )
2
(5, 3 , 2 )
(42 , 3, 22 )
(7, 2 , 1 )
4
(4 , 3, 1 )
(35 )
1
1
1
0
1
0
0
0
0
0
1
1
1
0
1
qqqqqq
>q
0
1
0
1
0
0
(3 , 1 )
0
0
0
0
0
0
0
0
3
(4 , 2 , 1 )
0
0
0
0
1
0
0
1
0
1
0
1
0
0
0
1
1
0
0
1
0
0
0
0
0
0
0
2
2
2
0
0
0
2
2
1
0
0
0
0
2
2
2
0
0
1
2
2
0
0
0
0
0
0
0
0
0
1
2
2
0
2
2
2
0
0
0
0
0
2
1
0
0
0
0
0
2
0
0
0
1
0
1
0
0
2
0
1
1
0
0
(3 , 1 )
0
0
2
0
0
0
2
1
0
1
1
0
0
1
2
1
1
0
0
0
1
0
0
1
0
0
2
2
0
0
2
0
0
1
1
1
0
0
1
1
0
1
0
0
2
1
0
0
1
1
(3, 1 )
(5, 2 , 1 )
0
2
0
0
0
1
1
0
1
1
(3 , 2 , 1 )
(3 , 2 , 1)
0
0
0
1
2
0
0
1
0
0
0
0
2
1
0
1
0
1
2
1
0
1
0
(3 , 2 , 1 )
0
0
2
0
0
0
1
0
1
1
1
1
0
0
2
1
0
0
0
1
0
0
2
0
2
1
0
0
(3, 2 , 1 )
2
0
0
0
0
2
2
2
1
0
0
(5, 3)
(5, 13 )
0
0
2
2
0
0
1
0
0
0
0
0
0
2
2
2
2
2
1
0
1
0
0
1
0
0
2
0
2
0
0
0
0
2
2
1
0
1
1
0
qqqqqq
>q qqqqqq
> q330
2
0
0
0
0
0
2
2
2
2
1
0
2
2
(3 , 1 )
0
qqqqqq
>q qqqqqq
> q334
(3, 22 , 1)
qqqqqq
>q qqqqqq
> q572
(3, 15 )
2
0
0
0
0
0
0
0
0
0
0
0
qqqqqq
>q
1
0
1
0
0
0
2
2
2
2
2
2
2
2
0
2
0
1
2
0
(24 )1
qqqqqq
>q
0
1
0
1
0
0
qqqqqq
>q
1
0
0
0
1
(24 )2
0
2
4
qqqqqq
>q
(2 , 1 )
qqqqqq
>q
(18 )
1
4
2
qqqqqq
>q qqqqqq
> q184
0
(3, 26 )
2
2
Ã` q
q
2
q
2
0
Ã` q
q
2
q
2
0
Ã` q
q
0
q
0
2
1
`Ãq q
q
1
0
0
`Ãq q
q
0
2
0
0
Ã` q
q
0
q
0
0
Ã` q
q
1
q
1
2
Ã` q
q
0
q
0
4
2
`Ãq q
q
2
8
2
`Ãq q
q
2
20
1
0
0
0
1
0
0
0
0
0
0
1
qqqqqq
>q
1
0
0
0
1
0
0
Diagram
Weight
2
(42 )2
0
5
2
so8
qqqqqq
>q qqqqqq
> q180
0
4
2
Weight
2
0
(42 )1
2
2
2
0
qqqqqq
>q qqqqqq
> q172
2
6
2
0
qqqqqq
>q qqqqqq
> q138
0
12
2
0
qqqqqq
>q qqqqqq
> q114
2
(32 , 19 )
0
0
0
(7, 1)
0
(5, 110 )
0
2
(3, 13 )
qqqqqq
>q qqqqqq
> q110
0
6
0
Diagram
0
Partition
2
(42 , 17 )
0
so6
qqqqqq
>q qqqqqq
> q110
2
(5, 3, 1 )
0
0
2
qqqqqq
>q qqqqqq
> q168
7
0
qqqqqq
>q qqqqqq
> q 94
2
(7, 18 )
0
qqqqqq
>q qqqqqq
> q910
(16 )
0
(5, 24 , 12 )
1
qqqqqq
>q
qqqqqq
>q qqqqqq
> q 90
2
2
0
qqqqqq
>q
(22 , 12 )
0
(5, 3, 22 , 13 )
0
qqqqqq
>q
qqqqqq
>q qqqqqq
> q100
3
4
1
1
2
1
0
qqqqqq
>q qqqqqq
> q 90
0
2
0
1
0
0
0
qqqqqq
>q qqqqqq
> q 86
0
2
1
0
0
1
1
qqqqqq
>q qqqqqq
> q 82
2
3
0
1
0
0
0
(32 )
0
4
2
1
0
0
0
1
qqqqqq
>q qqqqqq
> q 80
2
3
0
1
2
1
0
(5, 1)
2
(5, 32 , 14 )
2
2
0
0
0
0
qqqqqq
>q qqqqqq
> q 78
2
4
2
0
1
2
Partition
0
2
0
1
0
qqqqqq
>q qqqqqq
> q 74
2
2
0
0
qqqqqq
>q qqqqqq
> q 70
2
2
2
Weight
qqqqqq
>q qqqqqq
> q 68
0
(9, 16 )
8
qqqqqq
>q qqqqqq
> q 56
2
(7, 3, 22 , 1)
2
(continued)
Diagram
Partition
(3, 2 , 1 )
1
(5, 33 , 1)
Weight
qqqqqq
>q qqqqqq
> q 40
0
(5, 42 , 12 )
so15
(continued)
q Ãq
2
` q
q
0
q Ãq
2
` q
q
0
q Ãq
0
` q
q
2
q `Ãq q
0
q
2
0
q `Ãq q
2
q
0
q Ãq
0
` q
q
q Ãq
1
` q
q
6
1
1 0
q Ãq
1
` q
q
1
q Ãq
0
` q
q
2
q `Ãq q
0
q
2
0
q `Ãq q
2
q
0
q Ãq
0
` q
q
q Ãq
2
` q
q
2 24
2 0
q Ãq
0
` q
q
2
2
2
2
0
0
0
1
2
0
0
0
0
2
0
2
2
2
0
2
2
2
0
0
0
0
1
0
0
0
1
0
0
0
0
q Ãq
0
` q
q
0
0
0
q Ãq
0
` q
q
2
0
1
q Ãq
0
` q
q
4
0
0
q `Ãq q
0
q
4
2
0
q `Ãq q
2
q
4
0
0
q Ãq
1
` q
q
1 14
1
q Ãq
0 20
` q
q
0
2
q `Ãq q
0 20
q
2
2
q `Ãq q
2
q
0 20
2
q Ãq
2
` q
q
2 56
2
2
2
105
106
Experimental Mathematics, Vol. 11 (2002), No. 1
so10
Partition
Diagram
(9, 1)
so12
Weight
(7, 3)
q q `Ãq q
2
q q `Ãq q
0
q
q
0
2 0 0 0
0
2 2
(52 , 12 )
q q Ãq
2
q q Ãq
0
` q
q
` q
q
2
2
0
(5, 32 , 1)
q q Ãq
0
q q Ãq
0
` q
q
` q
q
4
0 2 0 0
0
2 2
(7, 22 , 1)
q q Ãq
2
q q Ãq
1
` q
q
` q
q
4
2
1
(42 , 3, 1)
q q `Ãq q
0
q q `Ãq q
0
q
q
6
0 1 0 1
0
0 2
(7, 15 )
q q `Ãq q
1
q q `Ãq q
1
q
q
10
1 0 1 0
1
2 0
(5, 3, 22 )
q q Ãq
0
q q Ãq
0
` q
q
` q
q
12
0
0
(42 , 22 )1
q q Ãq
1
q q Ãq
0
` q
q
` q
q
14
1 1 1 1
0
1 0
(42 , 22 )2
q q Ãq
0
q q Ãq
0
` q
q
` q
q
20
0
0
(5, 3, 14 )
q q `Ãq q
1
q q `Ãq q
1
q
q
20
1 0 2 0
1
1 0
(34 )
q q `Ãq q
0
q q `Ãq q
1
q
q
22
0
1
(42 , 14 )
q q Ãq
0
q q Ãq
1
` q
q
` q
q
30
0 1 1 1
1
0 1
(33 , 13 )
q q Ãq
1
q q Ãq
2
` q
q
` q
q
40
1
2
(5, 22 , 13 )
q q Ãq
0
q q Ãq
0
` q
q
` q
q
56
0
0
(32 , 22 , 12 )
q q `Ãq q
0
q q `Ãq q
2
q
q
60
0 2 2 0
2
1 0
(3, 24 , 1)
q q `Ãq q
0
q q `Ãq q
2 120
q
q
0
2
(5, 17 )
2
(7, 13 )
2
0
0
1
0
2
(52 )
0
(5, 3, 12 )
2
0
0
0
0
2
(42 , 12 )
0
(33 , 1)
0
(5, 22 , 1)
0
2
0
0
2
2
(5, 15 )
2
(32 , 22 )
2
0
2
2
0
0
(32 , 14 )
0
(3, 22 , 13 )
2
0
2
1
0
1
(24 , 12 )
0
(3, 17 )
2
(22 , 16 )
0
0
0
0
0
2
2
0
2
2
0
(110 )
0
0
0
2
2
2
(32 , 16 )
so12
Partition
(11, 1)
(9, 3)
(9, 13 )
(7, 5)
2
(7, 3, 1 )
2
(6 )1
2
(6 )2
(3, 22 , 15 )
Diagram
Weight
q q q `Ãq q
2
q
2
2
2
2
2
q q q Ãq
2
` q
q
2
2
2
0
2
q q q Ãq
0
` q
q
2
2
2
2
0
q q q Ãq
2
` q
q
2
0
2
0
2
q q q `Ãq q
0
q
2
2
0
2
0
q q q `Ãq q
0
q
0
0
(26 )2
q q q Ãq
0
` q
q
2
0
0 1 0 0
(24 , 14 )
q q q Ãq
0
` q
q
4
0
0
(3, 19 )
q q q Ãq
0
` q
q
4
0
0 0 1
(22 , 18 )
q q q `Ãq q
0
q
6
0
(112 )
2
0
0
0
0
0
1
0
1
0
q q q `Ãq q
2
q
0
q q q `Ãq q
0
q
6
2
0 0 0 0
q q q Ãq
0
` q
q
0
0
0
2
2
0
0
2
2
2
(26 )1
0
0
q q q Ãq
2
` q
q
6
0
0
0
Diagram
Partition
2
0
(continued)
Weight
q q q `Ãq q
0
q
1
q q q `Ãq q
0
q
0
q q q Ãq
1
` q
q
1
q q q Ãq
1
` q
q
0
q q q `Ãq q
0
q
2
q q q `Ãq q
1
q
0
q q q `Ãq q
2
q
0
q q q Ãq
0
` q
q
0
q q q Ãq
0
` q
q
2
q q q `Ãq q
0
q
0
q q q `Ãq q
0
q
2
q q q `Ãq q
0
q
2
q q q Ãq
0
` q
q
1
q q q Ãq
0
` q
q
0
q q q `Ãq q
1
q
1
q q q `Ãq q
0
q
2
q q q `Ãq q
0
q
2
q q q Ãq
0
` q
q
2
q q q Ãq
2
` q
q
0
q q q `Ãq q
0
q
0
q q q `Ãq q
0
q
2
q q q `Ãq q
0
q
2
0
2
2
0
2
2
0
0
2
0
0
0
2
0
1
2
0
1
0
0
0
2
2
0
2
1
2
0
2
2
0
0
2
0
1
1
0
2
2
0
0
0
0
0
0
0
1
1
2
1
0
0
2
0
0
2
0
0
0
0
0
1
0
0
0
0
2
2
0
0
0
0
0
0
0
2
1
0
1
1
0
0
0
0
0
0
1
0
0
0
1
1
0
1
0
2
0
0
0
0
0
0
1
0
0
0
0
2
0
0
q q q Ãq
0
` q
q
0
1
0
0
0
q q q Ãq
0
` q
q
0
0
0
0
0
q q q `Ãq q
1
q
8
q q q `Ãq q
0
q
12
q q q Ãq
0
` q
q
14
q q q Ãq
0
` q
q
16
q q q `Ãq q
0
q
20
q q q `Ãq q
0
q
20
q q q `Ãq q
0
q
22
q q q Ãq
2
` q
q
22
q q q Ãq
0
` q
q
22
q q q `Ãq q
1
q
24
q q q `Ãq q
1
q
26
q q q `Ãq q
0
q
28
q q q Ãq
0
` q
q
30
q q q Ãq
0
` q
q
40
q q q `Ãq q
0
q
54
q q q `Ãq q
0
q
56
q q q `Ãq q
1
q
58
q q q Ãq
1
` q
q
66
q q q Ãq
0
` q
q
70
q q q `Ãq q
2
q
70
q q q `Ãq q
2
q
76
q q q `Ãq q
0
q
120
0
0
1
0
2
2
2
2
1
1
0
0
1
2
1
2
2
1
2
2
0
2
0
2
1
0
0
0
0
0
0
1
1
0
1
0
1
2
1
1
0
0
2
2
0
0
0
2
0
1
0
0
1
0
0
2
1
2
2
0
0
1
2
2
0
2
1
0
0
0
0
0
2
0
0
1
1
0
0
0
0
0
1
1
2
0
2
0
q q q Ãq
2
` q
q
124
2?
2
2
2
0
q q q Ãq
2
` q
q
2
2
2
2
2
220
Chmutova, Ostrik: Calculating Canonical Distinguished Involutions in the Affine Weyl Groups
so14
Partition
(13, 1)
(11, 3)
(11, 13 )
(9, 5)
(72 )
(7, 32 , 1)
(62 , 12 )
(9, 22 , 1)
(52 , 3, 1)
(9, 15 )
2
(7, 3, 2 )
(52 , 22 )
(7, 3, 14 )
2
(5, 4 , 1)
(52 , 14 )
(5, 33 )
(5, 32 , 13 )
(42 , 32 )
3
(7, 2 , 1 )
(42 , 3, 13 )
(5, 3, 22 , 12 )
(34 , 12 )
(42 , 22 , 12 )
Weight
0
q q q q Ãq
2
` q
q
0
q q q q Ãq
0
` q
q
2
q q q q Ãq
2
` q
q
0
q q q q `Ãq q
0
q
1
q q q q `Ãq q
2
q
0
q q q q Ãq
0
` q
q
1
q q q q Ãq
0
` q
q
0
q q q q Ãq
1
` q
q
0
q q q q `Ãq q
1
q
1
q q q q `Ãq q
0
q
14
0
1 1 1 0 0
(5, 19 )
q q q q `Ãq q
0
q
0
q q q q `Ãq q
0
q
14
0
(32 , 18 )
q q q q Ãq
0
` q
q
2
q q q q Ãq
0
` q
q
20
0
2 0 0 0
(24 , 16 )
q q q q Ãq
0
` q
q
20
0
2
(3 , 2 , 1 )
q q q q Ãq
1
` q
q
22
1
(3, 24 , 13 )
2
2
2
2
2
0
2
0
2
2
2
2
2
2
2
0
2
2
2
2
2
2
2
2
0
2
0
2
0
0
2
0
2
2
2
2
2
0
2
0
0
2
1
0
2
2
0
2
0
2
0
2
2
0
2
2
0
2
0
2
0
0
1
0
2
0
0
0
q q q q `Ãq q
0
q
0
(5, 24 , 1)
q q q q Ãq
0
` q
q
2
(7, 17 )
q q q q Ãq
0
` q
q
4
(33 , 22 , 1)
q q q q Ãq
0
` q
q
4
(5, 3, 16 )
q q q q `Ãq q
0
q
6
(42 , 16 )
q q q q `Ãq q
1
q
6
(33 , 15 )
q q q q Ãq
0
` q
q
8
(5, 22 , 15 )
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
(32 , 24 )
q q q q Ãq
1
` q
q
12
1
(26 , 12 )
1
0
0
1
0
0
0
1
0
q q q q Ãq
1
` q
q
0
q q q q `Ãq q
0
q
0
q q q q `Ãq q
0
q
22
0
2 1 0 1 0
(3, 22 , 17 )
q q q q `Ãq q
1
q
1
q q q q `Ãq q
0
q
22
0
1
(3, 111 )
q q q q Ãq
0
` q
q
2
q q q q Ãq
1
` q
q
24
1
1 0 0 0
(22 , 110 )
q q q q Ãq
0
` q
q
0
q q q q Ãq
0
` q
q
24
0
(114 )
q q q q Ãq
0
` q
q
2
2
0
2
1
0
2
2
2
2
2
0
2
0
0
0
0
0
1
0
0
0
1
1
2
1
2
0
2
0
0
1
1
0
0
0
2
0
0
0
0
2
2
0
1
0
0
0
1
1
0
1
0
0
0
1
1
0
q q q q `Ãq q
0
q
0
q q q q `Ãq q
0
q
30
0
1 1 1 1 0
0
2
1
2
0
1
1
0
0
1
q q q q Ãq
0
` q
q
0
1
1
0
1
0
q q q q Ãq
0
` q
q
2
0
1
0
1
0
q q q q Ãq
0
` q
q
0
0
0
2
0
0
q q q q `Ãq q
1
q
0
2
0
0
0
1
q q q q `Ãq q
0
q
2
q q q q Ãq
0
` q
q
1
q q q q Ãq
0
` q
q
2
q q q q `Ãq q
0
q
2
q q q q `Ãq q
0
q
2
q q q q `Ãq q
0
q
2
0
2
0
0
2
4
1
2
0
0
2
0
1
0
2
1
2
0
2
0
0
0
0
0
1
0
1
q q q q `Ãq q
1
q
28
1
1
0
1
0
q q q q Ãq
0
` q
q
32
0
2
0
0
0
2
q q q q Ãq
0
` q
q
40
0
0
2
0
2
0
q q q q Ãq
1
` q
q
40
1
0 1 1 0
1
q q q q `Ãq q
1
q
42
1
0
2
0
1
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
q q q q `Ãq q
0
q
54
q q q q `Ãq q
0
q
56
q q q q Ãq
1
` q
q
56
q q q q Ãq
0
` q
q
58
q q q q `Ãq q
1
q
62
q q q q `Ãq q
0
q
64
q q q q `Ãq q
0
q
66
q q q q Ãq
1?
` q
q
70
q q q q Ãq
1
` q
q
0
q q q q Ãq
1
` q
q
0
0
0
1
0
0
0
0
0
1
0
1
0
2
2
0
0
0
0
0
0
q q q q `Ãq q
0
q
0
0
0
1
0
0
1
2
2
2
1
2
1
1
0
0
1
0
2
q q q q `Ãq q
0
q
0
0
2
1
2
0
1
0
1
0
1
2
0
0
0
1
0
2
1
0
0
1
0
1
0
0
1
0
q q q q Ãq
2?
` q
q
112
2
2
2
1
2
q q q q `Ãq q
0
q
0
0
2
0
q q q q `Ãq q
0
q
120
q q q q `Ãq q
1
q
122
2
2
2
2
2
1
0
0
0
1
q q q q `Ãq q
2?
q
140
2
2
2
0
2
0
q q q q Ãq
0
` q
q
0
1
0
1
0
0
q q q q Ãq
0
` q
q
1
0
0
0
0
1
q q q q `Ãq q
0
q
0
1
0
1
0
0
q q q q `Ãq q
0
q
2
0
0
0
0
0
q q q q `Ãq q
0
q
0
1
0
0
0
0
q q q q Ãq
0
` q
q
0
0
0
0
q q q q Ãq
0
` q
q
28
0
0 0 2 0
q q q q `Ãq q
1
q
1
2
1
2
q q q q Ãq
1
` q
q
Weight
q q q q `Ãq q
1
q
2
2
q q q q Ãq
0
` q
q
12
0
0 2 0 0
(continued)
Diagram
Partition
q q q q `Ãq q
2
q
2
0
(7, 5, 12 )
2
Diagram
2
(9, 3, 12 )
so14
0
0
q q q q `Ãq q
0
q
2
2
2
2
2
0
220
q q q q `Ãq q
2?
q
224
2
2 2 2 0
2
q q q q Ãq
2
` q
q
2
2
2
2
2
2
364
so16
Partition
(15, 1)
(13, 3)
(13, 13 )
(11, 5)
(11, 3, 12 )
Diagram
Weight
q q q q q `Ãq q
2
q q q q q `Ãq q
0 0
q
q
2 0 0 0 0 0 0
0
2 2 2 2 2
2
q q q q q Ãq
2
q q q q q Ãq
0 2
` q
q
` q
q
2
0
2
2
2
2
2
0
0
1
0
0
0
0
q q q q q Ãq
0
q q q q q Ãq
0
` q
q
` q
q
0 2 0 0 0 0 0
0 4
2 2 2 2 2
2
q q q q q Ãq
2
q q q q q Ãq
0 4
` q
q
` q
q
2
0
2
2
2
0
2
0
0
0
0
1
0
0
q q q q q `Ãq q
0
q q q q q `Ãq q
0 6
q
q
0
0
2
2
2
2
0
2
1
0
1
0
0
0
107
108
Experimental Mathematics, Vol. 11 (2002), No. 1
so16
Diagram
Partition
(9, 7)
2
(9, 5, 1 )
(82 )1
(82 )2
(72 , 12 )
(9, 32 , 1)
(11, 22 , 1)
(7, 5, 3, 1)
(62 , 3, 1)
(11, 15 )
(9, 3, 22 )
(53 , 1)
4
(9, 3, 1 )
(7, 5, 22 )
(7, 42 , 1)
(7, 5, 14 )
(7, 33 )
(62 , 22 )1
(62 , 22 )2
(52 , 32 )
(7, 32 , 13 )
2
4
(6 , 1 )
(52 , 3, 13 )
(9, 22 , 13 )
(5, 42 , 3)
so16
(continued)
Weight
0
2
0
2
0
0
0
0
0
0
1
(5, 42 , 13 )
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
8
0 1 0 0 0 1 0
0
2 2 0 2 0 2
(7, 3, 2 , 1 )
q q q q q Ãq
0
q q q q q Ãq
0
` q
q
` q
q
8
2
2
(5, 33 , 12 )
q q q q q Ãq
2
q q q q q Ãq
2
` q
q
` q
q
8
0 0 0 0 0 0 0
0
2 0 2 0 2
(44 )1
q q q q q `Ãq q
0
q q q q q `Ãq q
1
q
q
0
1 10
(44 )2
0
2
0
2
0
2
0
0
0
0
0
0
0
0
2
0
2
0
2
1
0
0
0
0
0
2
2
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
0 0 0 2 0 0 0
0 12
2 2 2 0 0 2
(52 , 22 , 12 )
q q q q q `Ãq q
1
q q q q q `Ãq q
0 14
q
q
1
0
(7, 24 , 1)
2
2
2
2
1
0
1
1
1
0
0
0
q q q q q Ãq
0
q q q q q Ãq
0 14
` q
q
` q
q
0 0 0 1 0 1 0
0
2 0 2 0 0 2
(9, 1 )
q q q q q Ãq
1
q q q q q Ãq
0 18
` q
q
` q
q
1
0
(5, 32 , 22 , 1)
0
2
0
1
1
0
0
0
0
1
0
1
q q q q q `Ãq q
0
q q q q q `Ãq q
0 20
q
q
0
0
2
2
2
2
2
0
2
2
0
0
0
0
7
(7, 3, 16 )
q q q q q `Ãq q
1
q q q q q `Ãq q
0
q
q
1 0 2 0 1 0 0
0 20
2 2 2 0 1 0
(42 , 3, 22 , 1)
q q q q q `Ãq q
0
q q q q q `Ãq q
0 20
q
q
0
0
(52 , 16 )
0
0
2
0
0
2
0
0
0
0
2
0
q q q q q Ãq
0
q q q q q Ãq
0 22
` q
q
` q
q
0 2 1 0 1 0 0
0
2 2 2 0 2 0
(3 , 1)
q q q q q Ãq
1
q q q q q Ãq
0
` q
q
` q
q
1
0 22
(5, 32 , 15 )
2
0
2
0
1
0
0
2
0
0
0
1
5
q q q q q `Ãq q
1
q q q q q `Ãq q
0 22
q
q
1 1 0 0 1 1 0
0
2 1 0 1 1 0
(7, 22 , 15 )
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
0
0 24
(5, 3, 24 )
q q q q q `Ãq q
0
q q q q q `Ãq q
0 24
q
q
0 0 1 1 0 1 0
0
2 0 0 0 2
(42 , 24 )1
q q q q q Ãq
0
q q q q q Ãq
0 24
` q
q
` q
q
2
2
(42 , 24 )2
q q q q q Ãq
2
q q q q q Ãq
2 24
` q
q
` q
q
0
0
(34 , 14 )
2
0
2
0
2
0
2
1
0
0
0
1
2
0
0
2
2
0
0
2
2
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
q q q q q `Ãq q
0
q q q q q `Ãq q
1
q
q
0 0 1 1 0 0 0
1 26
0 2 0 0 0 2
(42 , 22 , 14 )
q q q q q `Ãq q
0
q q q q q `Ãq q
0 28
q
q
0
0
(7, 19 )
q q q q q `Ãq q
0
q q q q q `Ãq q
1
q
q
0 2 0 1 0 0 0
1 28
0 2 0 2 0 1
8
(5, 3, 1 )
q q q q q Ãq
0
q q q q q Ãq
0
` q
q
` q
q
0
0 30
(42 , 18 )
q q q q q Ãq
0
q q q q q Ãq
0 30
` q
q
` q
q
0 1 1 1 1 0 0
0
2 2 1 0 1
(33 , 17 )
q q q q q `Ãq q
1
q q q q q `Ãq q
1
q
q
1
1 34
(28 )1
2
0
2
2
0
0
0
0
2
2
0
0
2
2
0
0
0
0
2
1
0
0
0
1
2
1
0
1
0
1
0
0
0
1
1
0
Diagram
Partition
q q q q q `Ãq q
2
q q q q q `Ãq q
0
q
q
6
2
0
2
0
(continued)
Weight
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
38
0
0
1
0
1
1
0
1
1
1
0
0
1
1
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
40
0
0
2 2 0 1 0 1
0 2 0 2 0 0
q q q q q Ãq
0
q q q q q Ãq
0
` q
q
` q
q
40
0
0
2
0
0
0
2
0
1
0
1
1
0
1
q q q q q Ãq
0
q q q q q Ãq
0
` q
q
` q
q
40
2
2
0 0 2 0 0
0 0 0 2 0 0
0
q q q q q `Ãq q
2
q q q q q `Ãq q
2
q
q
40
0
0
0
0
0
2
0
0
0
0
0
2
0
0
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
42
0
0
0 2 0 1 0 1
0 2 0 1 0 1
q q q q q `Ãq q
1
q q q q q `Ãq q
0
q
q
54
1
0
2
2
1
0
0
0
1
1
1
2
0
0
q q q q q Ãq
0
q q q q q Ãq
0
` q
q
` q
q
56
0
0
2 2 2 2 0 0
2 2 2 0 0 0
q q q q q Ãq
0
q q q q q Ãq
0
` q
q
` q
q
56
0
0
2
0
0
1
0
1
1
1
1
1
0
1
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
58
0
0
2
2
0
2
0
0
2
2
1
0
1
0
q q q q q `Ãq q
1
q q q q q `Ãq q
1
q
q
60
1
1
0 1 1 0 0 0
1 1 1 0 1 0
q q q q q `Ãq q
0
q q q q q `Ãq q
1
q
q
60
0
1
0
2
0
2
0
0
2
2
1
0
0
0
q q q q q Ãq
0
q q q q q Ãq
0
` q
q
` q
q
60
0
0
0 0 0 0 2 0
0 0 2 0 0 2
q q q q q Ãq
0
q q q q q Ãq
0
` q
q
` q
q
64
0
0
2
0
0
2
0
0
2
2
0
0
2
0
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
66
0
0
2 2 1 0 1 0
2 1 1 1 1 0
q q q q q `Ãq q
1
q q q q q `Ãq q
0
q
q
?
2
0
1
0
0
0
1
0
2
0
2
0
1
0
70
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
72
2
2
2 0 0 0 0
0 2 0 2 0 0
0
q q q q q Ãq
2
q q q q q Ãq
2
` q
q
` q
q
72
0
0
0
2
0
0
0
0
0
2
0
2
0
0
q q q q q Ãq
0
q q q q q Ãq
1
` q
q
` q
q
76
0
1
0
0
0
2
0
0
2
1
0
1
1
0
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
80
0
0
0 2 0 0 0 1
2 0 2 0 0 2
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
120
0
0
2
2
2
0
0
0
2
2
2
2
0
0
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
122
0
0
2 0 2 0 0 0
2 2 2 1 0 1
q q q q q Ãq
0
q q q q q Ãq
1
` q
q
` q
q
126
0
1
0
2
0
1
0
0
2
2
2
0
1
0
q q q q q Ãq
0
q q q q q Ãq
0
` q
q
` q
q
128
0
0
0 2 0 0 0
2 2 2 0 0 2
0
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
168
2
2?
0
0
0
0
0
0
0
2
0
2
0
2
Chmutova, Ostrik: Calculating Canonical Distinguished Involutions in the Affine Weyl Groups
so16
Diagram
Partition
(28 )2
(32 , 24 , 12 )
(32 , 22 , 16 )
(2 , 1 )
(22 , 112 )
(116 )
0
2
0
2
0
0
0
0
2
2
2
2
1
0
2
0
1
0
1
1
0
0
0
1
1
0
1
0
0
2
(32 )
1
0
1
0
1
1
1
0
1
0
1
1
0
0
0
0
0
1
1
0
1
0
0
0
1
0
0
0
0
0
1
0
1
0
0
1
0
0
1
0
0
0
0
0
2
0
2
0
qq
< q 34
0
2
3
0
qq
< q 56
0
2
2
2
1
0
0
0
0
0
0
1
0
0
0
1
2
0
0
2
1
0
0
0
0
0
2
0
1
0
2
2
0
0
0
0
2
0
2
0
1
qqq
< q qqq
< q 22
2
0
0
2
1
0
1
qqq
< q qqq
< q 34
1
0
0
2
3
0
0
qqq
< q qqq
< q 36
0
0
1
0
2
2
0
1
qqq
< q qqq
< q 54
0
0
1
0
0
2
3
0
1
qqq
< q qqq
< q 84
1
q q q q q `Ãq q
0
q q q q q `Ãq q
2
q
q
560
0
2
0 0 0 0 0
2 2 2 2 2 2
0
qqq
< q qqq
< q 20
2
0
0
0
2
1
1
0
0
0
qqq
< q qqq
< q 20
(4, 14 )
0
Weight
2
0
2
0
q q q q q Ãq
0
` q
q
0
0
1
1
qq
< q 20
qqq
< q qqq
< q 12
(32 , 12 )
(2, 16 )
0
0
2
0
2
0
0
2
(22 , 14 )
0
0
2
0
2
0
1
2
2
2
q q q q q Ãq
0
` q
q
0
0
qqq
< q qqq
< q 10
(32 , 2)
0
(23 , 12 )
0
2
qq
< q 10
qqq
< q qqq
<q 8
(6, 12 )
q q q q q `Ãq q
0
q
0
0
qqq
< q qqq
<q 4
2
0
0
0
2
0
q q q q q `Ãq q
0
q
1
1
qqq
< q qqq
<q 2
0
(24 )
0
0
0
qqq
< q qqq
<q 0
2
q q q q q `Ãq q
0
q
0
0
0
0
qq
<q 8
2
Diagram
2
(4, 2, 12 )
0
1
qq
<q
(8)
q q q q q Ãq
1
` q
q
1
0
qq
<q
Partition
q q q q q Ãq
0
` q
q
0
1
1
qq
<q 6
sp8
0
1
1
0
qq
<q
0
q q q q q `Ãq q
0
q
0
0
1
0
0
(4, 22 )
0
0
1
(42 )
0
2
qq
<q
2
q q q q q `Ãq q
0
q
1
2
0
qq
<q 2
qq
<q
0
(6, 2)
0
0
0
(23 )
q q q q q `Ãq q
0
q
0
0
0
qq
<q
(16 )
0
qq
<q 0
2
qq
<q
(2, 14 )
q q q q q Ãq
0
` q
q
0
2
0
0
Weight
qq
<q
2
2
(22 , 12 )
0
(33 , 22 , 13 )
8
2
q q q q q `Ãq q
0
q
0
0
(34 , 22 )
(26 , 14 )
0
q q q q q Ãq
0
` q
q
(42 , 3, 15 )
(3, 22 , 19 )
0
(4, 12 )
2
(4 , 3 , 1 )
(3, 2 , 1 )
0
Diagram
q q q q q `Ãq q
0
q
2
5
0
q q q q q Ãq
0
q q q q q Ãq
0
` q
q
` q
q
364
0
0
2 0 0 0 0 0
2 2 2 2 2 2
2
(5, 22 , 17 )
(3, 26 , 1)
0
q q q q q `Ãq q
0
q
(5, 24 , 13 )
4
q q q q q `Ãq q
0
q q q q q `Ãq q
0
q
q
220
0 2 2 2 2 2 0
0?
2 0 0 0 0
(4, 2)
0
q q q q q Ãq
0
q q q q q Ãq
1
` q
q
` q
q
222
0
1
(5, 3, 22 , 14 )
4
(6)
0
(3, 113 )
2
Partition
2
(32 , 110 )
2
Weight
q q q q q `Ãq q
2
q q q q q `Ãq q
2?
q
q
168
0
0
0
(5, 111 )
sp6
(continued)
(18 )
0
0
0
2
2
2
1
qqq
< q qqq
< q 120
0
0
0
0
2
2
2
2
0
sp10
sp4
Partition
Diagram
Partition
Weight
(10)
Diagram
qqqq
< q qqqq
<q 0
2
(4)
q
<q
2
(22 )
0
(2, 12 )
2
q
<q
1
(14 )
2
q
<q
0
q
<q
0
0
q
<q
0
(8, 2)
q
<q
2
(6, 4)
q
<q
10
(6, 22 )
q
<q
20
(52 )
0
0
2
2
0
2
2
2
2
0
0
0
0
0
2
2
0
2
0
1
0
0
0
qqqq
< q qqqq
<q 4
2
1
2
qqqq
< q qqqq
<q 2
2
1
Weight
0
2
0
2
0
0
0
1
0
qqqq
< q qqqq
<q 8
2
2
0
0
2
0
2
0
0
0
qqqq
< q qqqq
<q 8
0
2
0
2
0
1
0
0
0
1
109
110
Experimental Mathematics, Vol. 11 (2002), No. 1
sp10
Diagram
Partition
2
(4 , 2)
qqqq
< q qqqq
< q 10
(6, 32 )
qqqq
< q qqqq
< q 18
(6, 23 )
qqqq
< q qqqq
< q 20
(8, 2, 12 )
qqqq
< q qqqq
< q 20
(6, 4, 12 )
qqqq
< q qqqq
< q 22
(42 , 22 )
qqqq
< q qqqq
< q 24
(52 , 12 )
qqqq
< q qqqq
< q 34
(8, 14 )
qqqq
< q qqqq
< q 36
(6, 22 , 12 )
qqqq
< q qqqq
< q 40
(4, 22 )
qqqq
< q qqqq
< q 40
(34 )
qqqq
< q qqqq
< q 54
(4, 32 , 12 )
qqqq
< q qqqq
< q 58
(32 , 23 )
qqqq
< q qqqq
< q 60
(6, 2, 14 )
qqqq
< q qqqq
< q 82
(42 , 14 )
qqqq
< q qqqq
< q 84
(4, 32 , 2)
qqqq
< q qqqq
< q 120
(4, 23 , 12 )
qqqq
< q qqqq
< q 164
(32 , 22 , 12 )
qqqq
< q qqqq
< q 220
(26 )
1
(6, 2, 12 )
2
(4, 23 )
2
(42 , 12 )
0
(32 , 22 )
0
(6, 14 )
2
(4, 22 , 12 )
2
(32 , 2, 12 )
0
(25 )
0
(4, 2, 14 )
2
(32 , 14 )
0
(24 , 12 )
0
(23 , 14 )
0
(4, 16 )
2
(22 , 16 )
0
(2, 18 )
1
(110 )
3
0
2
2
0
2
0
2
1
2
0
1
0
0
2
0
0
1
1
0
0
0
2
1
0
0
0
0
1
0
1
0
1
0
0
1
0
0
0
0
0
1
1
1
0
1
1
0
1
0
0
0
0
1
0
0
0
0
0
2
0
0
0
2
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
2
1
2
0
2
0
2
2
2
0
2
2
2
2
2
2
2
2
1
0
1
0
0
1
2
0
2
0
1
0
1
1
3
0
2
0
1
1
2
0
3
0
2
1
2
0
3
0
2
2
2
2
2
2
2
2
1
0
1
0
1
1
0
0
1
0
2
1
0
2
2
1
2
3
2
0
Partition
(25 , 12 )
Weight
(12)
qqqqq
< q qqqqq
<q 0
qqqqq
< q qqqqq
<q 2
(32 , 16 )
qqqqq
< q qqqqq
<q 4
(4, 18 )
qqqqq
< q qqqqq
<q 6
(24 , 14 )
qqqqq
< q qqqqq
<q 8
(23 , 16 )
qqqqq
< q qqqqq
< q 10
(22 , 18 )
qqqqq
< q qqqqq
< q 10
(2, 110 )
qqqqq
< q qqqqq
< q 14
(112 )
2
(10, 2)
2
(8, 4)
2
(62 )
0
(8, 22 )
2
(10, 12 )
2
(6, 4, 2)
2
(52 , 2)
0
2
2
2
2
2
2
0
2
2
2
0
0
2
2
2
0
2
2
2
2
0
2
0
1
2
0
0
0
0
1
0
1
2
2
2
2
2
0
2
0
0
0
0
0
0
2
0
0
0
1
0
0
2
1
1
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
2
0
0
0
2
0
2
0
1
0
0
2
2
1
0
1
2
2
0
0
0
0
0
2
0
1
0
2
1
0
1
0
0
2
0
0
0
2
0
2
0
0
0
1
2
0
2
0
0
2
1
0
0
0
1
2
2
1
0
0
2
3
0
0
0
0
2
0
0
1
0
2
2
0
1
0
0
0
0
0
0
2
0
2
0
2
0
0
0
0
2
0
0
1
0
1
1
0
1
0
1
1
0
0
2
1
0
2
0
0
1
0
0
1
0
0
2
0
0
2
0
2
0
1
0
0
2
3
0
1
0
0
2
0
1
0
0
2
3
0
0
0
1
0
1
0
1
0
0
0
2
0
0
2
0
0
0
1
0
2
2
0
2
0
0
1
0
1
0
0
2
2
0
0
2
0
0
0
0
0
2
0
2
0
2
0
1
0
0
1
0
0
2
3
0
2
0
0
2
1
0
0
0
2
2
2
1
0
0
1
1
0
0
0
2
3
0
0
2
0
qqqqq
< q qqqqq
< q 94
0
(4, 2, 16 )
0
qqqqq
< q qqqqq
< q 86
0
Diagram
1
qqqqq
< q qqqqq
< q 84
2
(32 , 2, 14 )
sp12
1
qqqqq
< q qqqqq
< q 82
2
(6, 16 )
0
qqqqq
< q qqqqq
< q 70
0
(4, 22 , 14 )
1
qqqqq
< q qqqqq
< q 64
0
2
0
qqqqq
< q qqqqq
< q 60
2
0
1
qqqqq
< q qqqqq
< q 60
1
0
1
qqqqq
< q qqqqq
< q 56
0
0
0
qqqqq
< q qqqqq
< q 54
2
0
1
qqqqq
< q qqqqq
< q 44
0
0
0
qqqqq
< q qqqqq
< q 42
1
1
0
qqqqq
< q qqqqq
< q 40
0
0
2
qqqqq
< q qqqqq
< q 40
2
0
0
qqqqq
< q qqqqq
< q 36
2
1
0
qqqqq
< q qqqqq
< q 34
2
0
0
qqqqq
< q qqqqq
< q 24
0
0
2
qqqqq
< q qqqqq
< q 22
0
1
0
qqqqq
< q qqqqq
< q 22
2
0
0
qqqqq
< q qqqqq
< q 20
2
0
2
qqqqq
< q qqqqq
< q 20
2
0
0
qqqqq
< q qqqqq
< q 18
2
0
Weight
qqqqq
< q qqqqq
< q 16
0
0
(continued)
Diagram
Partition
(4 )
2
(4, 32 )
Weight
qqqq
< q qqqq
< q 10
0
(8, 12 )
sp12
(continued)
0
0
0
1
0
2
2
0
2
0
1
qqqqq
< q qqqqq
< q 120
2
0
1
0
0
0
2
2
2
2
0
0
qqqqq
< q qqqqq
< q 122
0
2
0
0
0
0
2
2
2
1
0
1
qqqqq
< q qqqqq
< q 164
2
1
0
0
0
0
2
2
2
3
0
0
qqqqq
<q
0
0
0
1
0
0
qqqqq
<q
0
0
1
0
0
0
qqqqq
<q
0
1
0
0
0
0
qqqqq
<q
1
0
0
0
0
0
qqqqq
< q qqqqq
< q 364
0
0
0
0
0
0
2
2
2
2
2
2
Chmutova, Ostrik: Calculating Canonical Distinguished Involutions in the Affine Weyl Groups
sp14
Partition
(14)
Diagram
(52 , 14 )
qqqqqq
< q qqqqqq
<q 4
(34 , 2)
qqqqqq
< q qqqqqq
<q 6
(42 , 22 , 12 )
qqqqqq
< q qqqqqq
<q 8
(4, 32 , 2, 12 )
qqqqqq
< q qqqqqq
< q 10
(4, 25 )
qqqqqq
< q qqqqqq
< q 10
(34 , 12 )
qqqqqq
< q qqqqqq
< q 10
(6, 22 , 14 )
qqqqqq
< q qqqqqq
< q 12
(42 , 2, 14 )
qqqqqq
< q qqqqqq
< q 16
(8, 16 )
qqqqqq
< q qqqqqq
< q 18
(4, 32 , 14 )
qqqqqq
< q qqqqqq
< q 20
(4, 24 , 12 )
qqqqqq
< q qqqqqq
< q 20
(6, 2, 16 )
qqqqqq
< q qqqqqq
< q 20
(27 )
qqqqqq
< q qqqqqq
< q 22
(42 , 16 )
qqqqqq
< q qqqqqq
< q 22
(6, 18 )
qqqqqq
< q qqqqqq
< q 24
(32 , 18 )
qqqqqq
< q qqqqqq
< q 26
(4, 23 , 14 )
qqqqqq
< q qqqqqq
< q 28
(4, 22 , 16 )
qqqqqq
< q qqqqqq
< q 30
(4, 2, 18 )
qqqqqq
< q qqqqqq
< q 34
(4, 11 0)
qqqqqq
< q qqqqqq
< q 36
(32 , 24 )
qqqqqq
< q qqqqqq
< q 38
(32 , 23 , 12 )
qqqqqq
< q qqqqqq
< q 40
(32 , 22 , 14 )
qqqqqq
< q qqqqqq
< q 40
(32 , 2, 16 )
qqqqqq
< q qqqqqq
< q 42
(26 , 12 )
qqqqqq
< q qqqqqq
< q 42
(25 , 14 )
qqqqqq
< q qqqqqq
< q 42
(24 , 16 )
qqqqqq
< q qqqqqq
< q 44
(23 , 18 )
qqqqqq
< q qqqqqq
< q 50
(22 , 110 )
qqqqqq
< q qqqqqq
< q 54
(2, 112 )
qqqqqq
< q qqqqqq
< q 56
(114 )
2
(10, 22 )
2
(12, 12 )
2
(8, 4, 2)
2
(72 )
0
(62 , 2)
0
(6, 42 )
2
(8, 32 )
2
(10, 2, 12 )
2
(8, 23 )
2
(52 , 4)
0
(8, 4, 12 )
2
(6, 4, 22 )
2
(62 , 12 )
0
(52 , 22 )
0
(43 , 2)
0
(6, 32 , 2)
2
(10, 14 )
2
(8, 22 , 12 )
2
(6, 4, 2, 12 )
2
(6, 24 )
2
(42 , 32 )
0
(42 , 23 )
0
(6, 32 , 12 )
2
(52 , 2, 12 )
0
(43 , 12 )
0
(4, 32 , 22 )
1
(8, 2, 14 )
2
(6, 4, 14 )
2
qqqqqq
< q qqqqqq
<q 2
2
(8, 6)
3
2
2
2
2
0
2
2
2
2
2
0
2
2
2
1
2
0
2
2
0
1
2
2
0
2
1
2
1
2
0
0
2
0
2
2
2
2
2
2
0
0
0
0
1
2
2
1
0
2
0
0
2
0
2
2
2
0
0
0
0
0
2
1
2
2
2
2
0
0
2
2
2
2
2
2
0
2
0
0
2
0
2
1
0
1
2
0
0
0
1
0
1
1
0
0
0
0
2
2
2
2
0
2
0
0
0
0
1
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
1
1
0
0
1
1
2
0
0
0
0
1
0
2
0
0
1
1
0
1
1
0
1
1
0
1
0
1
1
0
1
0
0
0
1
1
0
0
2
2
2
2
2
0
2
0
2
2
0
0
2
0
0
2
0
0
2
0
0
0
0
2
0
2
0
0
0
0
0
0
0
0
0
0
0
2
0
1
0
0
1
2
0
0
2
0
2
0
0
0
2
2
2
0
1
0
2
2
2
0
2
2
0
1
0
0
2
1
1
0
1
0
0
2
2
0
1
2
1
1
1
0
3
2
2
2
0
2
1
1
1
1
3
3
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
2
0
0
0
0
1
0
0
1
0
1
0
0
0
0
1
0
0
0
1
0
0
2
1
0
1
0
1
0
0
0
1
0
0
1
0
2
1
1
2
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
1
0
1
1
0
0
1
0
1
1
0
0
1
1
0
1
0
0
1
1
0
0
0
0
2
0
0
2
0
2
0
0
0
1
0
2
2
0
1
0
1
0
0
1
0
1
0
0
2
1
1
0
1
1
0
0
0
0
0
0
2
0
2
0
2
0
1
0
0
0
2
0
0
0
2
1
0
1
1
0
1
2
0
0
1
0
0
2
3
0
2
0
0
0
2
0
0
1
0
0
2
3
0
1
0
1
0
2
2
1
0
0
0
2
2
2
1
0
0
0
0
1
1
0
0
0
2
2
1
0
1
1
0
0
0
0
0
1
0
2
2
0
2
0
1
0
2
0
1
0
0
0
2
2
2
2
0
0
0
0
0
0
0
0
2
0
2
0
2
0
2
0
2
0
1
0
0
0
2
2
2
1
0
1
0
2
1
0
0
0
0
2
2
2
3
0
0
0
2
0
0
0
0
0
2
2
2
2
1
0
1
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
1
0
0
0
1
0
1
0
0
1
0
0
1
0
1
0
0
0
1
1
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
qqqqqq
<q
0
0
0
qqqqqq
<q
0
0
0
qqqqqq
<q
0
0
1
qqqqqq
<q
0
1
0
qqqqqq
<q
0
0
0
qqqqqq
<q
0
0
0
qqqqqq
<q
0
0
1
qqqqqq
<q
0
0
2
qqqqqq
<q
0
0
2
qqqqqq
<q
2
0
0
qqqqqq
<q
2
0
0
qqqqqq
<q
2
0
0
qqqqqq
<q
2
0
2
qqqqqq
< q qqqqqq
< q 222
0
1
0
qqqqqq
< q qqqqqq
< q 164
2
0
2
qqqqqq
< q qqqqqq
< q 122
0
0
0
qqqqqq
< q qqqqqq
< q 112
0
0
0
qqqqqq
< q qqqqqq
< q 120
2
0
0
qqqqqq
< q qqqqqq
< q 94
2
0
2
qqqqqq
< q qqqqqq
< q 92
1
0
0
qqqqqq
< q qqqqqq
< q 84
2
0
2
qqqqqq
< q qqqqqq
< q 84
0
0
2
qqqqqq
< q qqqqqq
< q 82
2
0
0
qqqqqq
< q qqqqqq
< q 76
0
1
1
qqqqqq
< q qqqqqq
< q 70
2
0
0
qqqqqq
< q qqqqqq
< q 70
1
0
0
qqqqqq
< q qqqqqq
< q 62
0
0
0
qqqqqq
< q qqqqqq
< q 60
0
0
2
qqqqqq
< q qqqqqq
< q 60
0
0
Weight
qqqqqq
< q qqqqqq
< q 60
2
0
(continued)
Diagram
Partition
(6, 2 , 1 )
2
(10, 4)
Weight
qqqqqq
< q qqqqqq
<q 0
2
(12, 2)
sp14
111
1
0
0
0
0
0
qqqqqq
<q
1
0
0
0
0
0
0
qqqqqq
< q qqqqqq
< q 560
0
0
0
0
0
0
0
2
2
2
2
2
2
2
112
Experimental Mathematics, Vol. 11 (2002), No. 1
The Exceptional Groups
G2
Diagram
Weight
2
0
2
1
0
3
0
q< q 56
q< q
0
q
q
2
0
qqqqq qqqqq 0
2
2
2
2
2
0
0
0
0
q
q
0
1
q q q q q q q q q q 42
0
0
2
q
q
2
1
qqqqq qqqqq 2
2
2
0
2
2
0
0
0
0
0
2
2
Weight
qq
>q q q q
>q q 0
2
2
0
0
0
2
0
2
0
0
0
1
qq
>q q q q
>q q 6
0
2
0
2
0
0
1
2
0
0
0
0
0
2
qq
>q q q q
>q q 12
0
2
0
0
0
1
0
0
1
2
2
0
0
0
qq
>q q q q
>q q 28
1
0
1
0
1
1
0
1
0
1
0
1
1
0
qq
>q q q q
>q q 34
2
0
0
1
1
0
0
3
qq
>q q q q
>q q 38
0
0
1
0
0
0
2
1
qq
>q q q q
>q q 56
2
0
0
0
0
0
2
2
qq
>q q q q
>q q 70
0
0
0
2
2
1
0
1
qq
>q q q q
>q q 72
0
1
0
0
2
0
0
2
0
2
1
0
0
0
1
2
0
2
0
2
0
0
1
0
0
0
1
2
1
0
0
3
qq
>q q q q
>q q 156
1
0
0
0
1
1
0
0
0
1
1
2
2
0
0
2
0
0
2
0
0
1
1
0
1
1
1
0
0
0
2
2
0
0
0
1
1
1
0
1
1
q
q
2
1
q q q q q q q q q q 70
0
q
q
2
1
q q q q q q q q q q 10
0
0
0
0
2
1
0
1
2
q
q
0
1
q q q q q q q q q q 78
1
0
q
q
1
0
q q q q q q q q q q 12
1
1
0
1
1
0
1
0
1
0
1
0
0
1
1
1
1
1
q
q
0
2
q q q q q q q q q q 120
0
1
q
q
1
1
q q q q q q q q q q 14
2
1
0
1
2
0
0
1
0
0
0
0
1
2
0
2
0
2
q
q
1
2
q q q q q q q q q q 168
0
0
q
q
2
0
q q q q q q q q q q 16
0
0
2
0
0
2
0
0
0
0
0
0
0
2
2
0
2
2
q
q
0
2
q q q q q q q q q q 312
2
0
0
0
0
0
2
2
2
2
2
2
2
E7
q
q
0
0
q q q q q q q q q q 18
0
0
2
0
0
1
0
1
0
1
q
q
2
2
q q q q q q q q q q 20
2
0
0
0
2
1
0
0
0
1
Diagram
0
1
0
1
0
1
0
1
0
1
1
0
1
0
1
0
1
1
1
0
1
0
0
0
1
2
0
0
0
2
2
2
2
2
0
0
0
0
0
0
2
0
2
2
2
1
0
0
0
0
0
q
q
2
0
qqqqqq qqqqqq 4
2
q
q
2
2
q q q q q q q q q q 40
2
q
q
2
0
qqqqqq qqqqqq 2
2
q
q
0
0
q q q q q q q q q q 34
Weight
q
q
2
0
qqqqqq qqqqqq 0
2
q
q
1
1
q q q q q q q q q q 30
qq
>q q q q
>q q 312
0
0
1
0
qq
>q q q q
>q q 118
0
0
0
qq
>q q q q
>q q 34
0
1
0
qq
>q q q q
>q q 16
1
1
0
qq
>q q q q
>q q 8
2
0
0
qq
>q q q q
>q q 2
2
1
q
q
1
2
q q q q q q q q q q 60
q
q
0
0
qqqqq qqqqq 6
F4
2
0
2
2
2
1
q
q
0
2
q q q q q q q q q q 56
q
q
2
0
qqqqq qqqqq 4
Diagram
Weight
Diagram
1
q< q 18
1
0
0
1
q< q
0
Weight
(continued)
q< q 14
q< q
1
0
q< q 2
q< q
0
Diagram
E6
q< q 0
q< q
2
E6
2
0
2
0
2
0
0
0
0
1
0
q
q
0
0
qqqqqq qqqqqq 6
2
0
2
0
2
2
0
1
0
0
0
0
Chmutova, Ostrik: Calculating Canonical Distinguished Involutions in the Affine Weyl Groups
E7
(continued)
E7
(continued)
Diagram
Weight
Diagram
Weight
q
q
0
0
qqqqqq qqqqqq 6
2
2
2
0
2
0
0
0
0
0
0
2
q
q
0
1
qqqqqq qqqqqq 8
2
0
2
0
2
0
0
0
0
0
0
1
q
q
0
0
q q q q q q q q q q q q 12
2
0
2
0
0
2
0
0
1
0
0
0
q
q
1
0
q q q q q q q q q q q q 14
2
1
0
1
2
2
1
1
0
0
0
0
q
q
0
2
q q q q q q q q q q q q 14
0
0
2
0
2
0
0
0
0
0
0
0
q
q
1
0
q q q q q q q q q q q q 16
2
1
0
1
1
0
0
0
0
0
2
0
q
q
0
0
q q q q q q q q q q q q 18
0
0
2
0
0
2
0
1
0
0
1
0
q
q
1
0
q q q q q q q q q q q q 20
2
1
0
1
0
2
2
0
0
0
1
0
q
q
0
0
q q q q q q q q q q q q 22
2
2
0
0
2
0
2
0
0
0
0
2
q
q
0
0
q q q q q q q q q q q q 24
0
2
0
0
2
0
1
0
0
1
0
1
q
q
0
0
q q q q q q q q q q q q 28
2
0
0
2
0
0
0
0
1
0
1
0
q
q
1
0
q q q q q q q q q q q q 30
0
1
0
1
0
2
1
1
0
0
1
0
q
q
0
0
q q q q q q q q q q q q 30
0
0
2
0
0
0
0
0
0
2
0
0
q
q
0
1
q q q q q q q q q q q q 32
1
0
1
0
2
0
1
0
0
1
0
0
q
q
0
0
q q q q q q q q q q q q 34
1
0
1
0
1
2
0
1
1
0
0
0
q
q
0
0
q q q q q q q q q q q q 40
2
0
1
0
1
0
2
0
0
0
2
0
q
q
0
0
q q q q q q q q q q q q 42
1
0
1
0
1
0
1
0
1
0
1
0
q
q
0
0
q q q q q q q q q q q q 48
0
0
0
2
0
0
0
0
2
0
0
0
q
q
1
0
q q q q q q q q q q q q 54
2
1
0
0
0
1
1
1
0
0
2
0
q
q
0
0
q q q q q q q q q q q q 56
2
0
0
0
2
2
2
2
0
0
0
0
q
q
0
1
q q q q q q q q q q q q 58
2
0
0
0
2
0
2
1
0
0
0
1
q
q
0
0
q q q q q q q q q q q q 62
0
0
1
0
1
0
2
0
0
2
0
0
q
q
1
0
q q q q q q q q q q q q 70
0
1
0
0
0
1
1
0
1
0
2
0
q
q
0
0
q q q q q q q q q q q q 70
2
2
0
0
0
0
2
0
0
0
2
2
q
q
0
0
q q q q q q q q q q q q 72
0
2
0
0
0
0
1
0
1
0
1
2
q
q
0
0
q q q q q q q q q q q q 78
1
0
0
1
0
1
1
1
1
0
1
0
q
q
0
1
q q q q q q q q q q q q 84
1
0
1
0
0
0
1
0
1
0
1
1
q
q
0
0
q q q q q q q q q q q q 88
0
1
0
0
1
0
0
1
1
1
0
1
113
114
Experimental Mathematics, Vol. 11 (2002), No. 1
E7
E8
(continued)
Diagram
Weight
Diagram
q
q
2
0
q q q q q q q q q q q q 112
0
0
0
0
0
0
0
0
2
0
2
0
q
q
0
0
q q q q q q q q q q q q 120
2
0
0
0
0
2
2
2
0
0
2
0
q
q
0
1
q q q q q q q q q q q q 122
0
0
0
0
2
0
2
1
0
1
1
0
q
q
0
1
q q q q q q q q q q q q 124
2
0
0
0
1
0
2
1
0
1
0
2
q
q
0
0
q q q q q q q q q q q q 126
0
0
1
0
0
0
2
0
2
0
0
2
q
q
0
1
q q q q q q q q q q q q 166
1
0
0
0
1
0
2
1
0
1
2
0
q
q
1
0
qqqqqq qqqqqq
? 168
0
0
0
0
0
1
2
0
2
0
2
0
q
q
0
1
q q q q q q q q q q q q 220
2
0
0
0
0
0
2
1
0
1
2
2
q
q
0
0
qqqqqq qqqqqq
? 222
0
1
0
0
0
0
2
0
2
0
2
2
q
q
0
0
q q q q q q q q q q q q 312
0
0
0
0
0
2
2
2
2
0
2
0
q
q
0
2
qqqqqq qqqqqq
? 318
0
0
0
0
1
0
2
2
0
2
0
2
q
q
0
2
qqqqqq qqqqqq
? 462
1
0
0
0
0
0
2
2
0
2
2
2
q
q
0
2
q q q q q q q q q q q q 798
0
0
0
0
0
0
2
2
2
2
2
2
Weight
q
q
2
0
qqqqqqq qqqqqqq 0
2
2
2
2
2
2
2
0
0
0
0
0
0
0
q
q
2
0
qqqqqqq qqqqqqq 2
2
2
0
2
2
2
2
0
0
0
0
0
0
1
q
q
2
0
qqqqqqq qqqqqqq 4
2
2
0
2
0
2
2
1
0
0
0
0
0
0
q
q
0
0
qqqqqqq qqqqqqq 6
2
0
2
0
2
2
2
0
0
0
0
0
1
0
q
q
0
1
qqqqqqq qqqqqqq 8
2
0
2
0
2
0
2
0
0
0
0
0
0
0
q
q
0
0
q q q q q q q q q q q q q q 12
2
0
2
0
0
2
2
0
0
0
0
1
0
0
q
q
1
0
q q q q q q q q q q q q q q 14
2
1
0
1
2
2
2
0
0
0
0
0
1
1
q
q
0
0
q q q q q q q q q q q q q q 14
2
0
2
0
0
2
0
0
1
0
0
0
0
0
q
q
1
0
q q q q q q q q q q q q q q 20
2
1
0
1
0
2
2
1
0
0
0
0
0
2
q
q
1
1
q q q q q q q q q q q q q q 22
2
1
0
1
1
0
1
1
0
0
0
0
0
0
q
q
0
0
q q q q q q q q q q q q q q 28
2
0
0
2
0
0
2
1
0
0
0
1
0
0
q
q
1
0
q q q q q q q q q q q q q q 30
0
1
0
1
0
2
2
1
0
0
0
0
1
1
q
q
0
0
q q q q q q q q q q q q q q 40
2
0
1
0
1
0
2
2
0
0
0
0
0
2
q
q
0
0
q q q q q q q q q q q q q q 42
1
0
1
0
1
0
2
1
0
0
0
1
0
1
Chmutova, Ostrik: Calculating Canonical Distinguished Involutions in the Affine Weyl Groups
E8
E8
(continued)
Weight
Diagram
0
0
2
0
0
2
0
0
0
0
2
0
0
q
q
0
0
q q q q q q q q q q q q q q 50
0
0
2
0
0
2
2
0
1
0
0
1
0
0
q
q
1
0
q q q q q q q q q q q q q q 54
2
1
0
0
0
1
2
2
0
0
0
0
1
1
q
0
qqqqqqq
1
0
1
0
1
1
0
1
0
1
0
0
0
0
0
2
2
0
0
0
2
0
2
0
0
0
0
0
2
2
0
0
0
0
0
1
2
q
q
0
0
q q q q q q q q q q q q q q 60
0
0
0
2
0
0
0
0
0
1
0
0
1
0
q
q
0
2
q q q q q q q q q q q q q q 64
2
0
0
0
2
0
0
0
0
0
0
0
0
2
q
q
0
0
q q q q q q q q q q q q q q 78
1
0
0
1
0
1
2
q
0
qqqqqqq
0
0
1
0
1
0
0
q
0
qqqqqqq
1
0
0
1
0
1
0
q
1
qqqqqqq
0
1
0
0
0
1
0
q
0
qqqqqqq
0
1
0
0
1
0
1
1
0
0
0
1
q
q
0
0
qqqqqqq qqqqqqq
?
0
0
1
0
0
1
0
0
0
0
2
0
0
80
0
q
q
0
1
q q q q q q q q q q q q q q 82
0
1
q
q
0
1
q q q q q q q q q q q q q q 58
2
0
0
1
0
1
0
2
1
0
0
1
0
0
0
0
2
0
0
2
0
0
1
0
0
0
1
1
q
0
qqqqqqq
1
2
1
0
q
q
0
0
q q q q q q q q q q q q q q 56
2
q
0
qqqqqqq
q
q
0
1
q q q q q q q q q q q q q q 98
q
0
qqqqqqq
1
Weight
Diagram
q
q
0
0
q q q q q q q q q q q q q q 48
0
(continued)
1
0
0
0
1
1
1
0
1
0
1
0
0
q
q
1
0
q q q q q q q q q q q q q q 116
0
1
0
0
0
1
2
1
0
1
0
0
1
1
q
q
0
0
q q q q q q q q q q q q q q 174
0
0
2
0
0
0
2
0
0
2
0
0
1
0
q
q
2
0
q q q q q q q q q q q q q q 112
0
0
0
0
0
0
2
2
0
0
0
2
0
0
q
q
0
0
q q q q q q q q q q q q q q 120
2
0
0
0
0
2
2
2
0
0
0
0
2
2
q
q
0
0
q q q q q q q q q q q q q q 122
2
0
0
0
0
2
0
1
0
0
1
0
1
2
q
0
qqqqqqq
0
1
0
0
1
0
0
q
q
0
0
q q q q q q q q q q q q q q 126
0
0
1
0
0
0
2
0
0
1
0
1
0
2
q
0
qqqqqqq
2
0
0
0
1
0
1
q
q
0
0
q q q q q q q q q q q q q q 128
0
0
0
0
2
0
0
0
0
0
2
0
0
2
115
116
Experimental Mathematics, Vol. 11 (2002), No. 1
E8
E8
(continued)
Weight
Diagram
0
1
0
0
0
q
q
0
1
q q q q q q q q q q q q q q 330
1
1
q
0
qqqqqqq
1
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
1
1
0
1
0
1
1
0
q
q
1
0
qqqqqqq qqqqqqq
? 168
0
0
0
0
0
1
2
2
0
0
0
2
0
2
q
0
qqqqqqq
0
0
1
0
0
0
0
0
0
0
0
0
2
2
1
0
0
0
1
2
q
q
0
2
q q q q q q q q q q q q q q 230
1
0
1
0
1
2
2
1
0
1
0
0
0
2
q
0
qqqqqqq
1
0
0
0
1
0
0
0
0
0
1
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
2
0
0
0
0
2
0
0
1
1
2
0
0
0
1
0
1
0
0
0
0
1
0
0
0
0
0
0
0
1
1
0
1
1
1
0
q
q
0
1
qqqqqqq qqqqqqq
?
462
q
q
0
0
qqqqqqq qqqqqqq
?
464
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
2
0
2
2
2
0
1
0
2
0
2
0
1
0
0
0
0
2
2
2
0
2
2
q
0
qqqqqqq
1
0
0
0
0
0
1
0
0
0
0
0
0
2
0
2
0
2
0
560
2
0
0
0
0
0
2
2
1
0
1
2
2
2
0
0
0
0
1
0
2
0
2
0
2
2
800
2
q
q
0
2
qqqqqqq qqqqqqq
? 1040
0
0
1
368
0
0
1
2
0
0
0
2
2
2
q
q
0
0
q q q q q q q q q q q q q q 314
0
2
q
q
0
0
qqqqqqq qqqqqqq
?
1
2
2
q
q
0
0
qqqqqqq qqqqqqq
?
0
q
q
0
0
q q q q q q q q q q q q q q 312
0
0
q
q
0
1
q q q q q q q q q q q q q q 798
q
0
qqqqqqq
0
2
0
q
0
qqqqqqq
0
1
q
q
1
0
qqqqqqq qqqqqqq
?
q
1
qqqqqqq
0
0
q
q
0
1
q q q q q q q q q q q q q q 336
1
q
q
0
1
q q q q q q q q q q q q q q 220
2
0
q
0
qqqqqqq
q
q
2
0
q q q q q q q q q q q q q q 168
0
0
q
0
qqqqqqq
q
0
qqqqqqq
1
Weight
Diagram
q
0
qqqqqqq
0
(continued)
1
0
1
0
1
2
2
0
0
0
0
0
0
2
2
0
2
0
2
2
q
q
0
2
qqqqqqq qqqqqqq
? 1520
0
0
0
0
0
0
1
2
2
0
2
2
2
2
q
q
0
2
q q q q q q q q q q q q q q 2480
0
0
0
0
0
0
0
2
2
2
2
2
2
2
Chmutova, Ostrik: Calculating Canonical Distinguished Involutions in the Affine Weyl Groups
ACKNOWLEDGEMENTS
We would like to thank David Vogan and Pramod Achar for
useful conversations. We are grateful to the referee for carefully reading this paper and answering an open question posed
in an earlier version.
REFERENCES
117
[Lusztig 89] G. Lusztig. “Cells in affine Weyl groups IV.” J.
Fac. Sci. Univ. Tokyo Sect. IA Math. 36:2 (1989), 297—
328.
[McGovern 89] W. McGovern. “Rings of regular functions on
nilpotent orbits and their covers.” Invent. Math. 97: 1
(1989), 209—217.
[Bezrukavnikov 00] R. Bezrukavnikov. “On tensor categories
attached to cells in affine Weyl groups.” preprint
RT/0010089.
[Ostrik 00] V. Ostrik. “On the equivariant K−theory of the
nilpotent cone.” Representation Theory 4 (2000), 296—
305.
[Bezrukavnikov 01] R. Bezrukavnikov. “Quasi-exceptional
sets and equivariant coherent sheaves on the nilpotent
cone.” preprint RT/0102039.
[Panyushev 91] Panyushev, “Rationality of singularities and
the Gorenstein property of nilpotent orbits.” Funct.
Anal. Appl. 25: 3 (1991), 225—226.
[Collingwood, McGovern 93] D. Collingwood and W. McGovern. Nilpotent orbits in semisimple Lie algebras,
Van Nostrand Reinhold Co., New York, 1993.
[Hinich 91] V. Hinich. “On the singularities of nilpotent orbits.” Israel J. Math. 73: 3 (1991), 297—308.
[Kostant 63] B. Kostant. “Lie group representations on polynomial rings.” Amer. J. Math. 85: (1963), 327—404.
[Lusztig 87] G. Lusztig. “Cells in affine Weyl groups II.” J.
Algebra 109: 2 (1987), 223—243.
[Lusztig 88] G. Lusztig and N. Xi. “Canonical left cells in
affine Weyl groups.” Adv. in Math. 72:2 (1988), 284—
288.
[Spaltenstein 82] N. Spaltenstein. Classes unipotentes et
sous-groupes de Borel., Lecture Notes in Math. 946,
Berlin-Heidelbeg, Springer-Verlag, 1982.
[Vogan 00] D. Vogan. “The method of coadjoint orbits for
real reductive groups.” In Representation theory of Lie
groups (Park City, UT, 1998), IAS/Park City Math.
Ser., 8, pp. 179—238, Amer. Math. Soc., Providence,
RI, 2000.
[Xi, 00] N. Xi. “The based ring of two-sided cells of affine
Weyl groups of type Ãn−1 .” preprint QA/0010159.
Tanya Chmutova, Department of Mathematics, 1 Oxford St, Cambridge, MA 02138
(chmutova@math.harvard.edu)
Viktor Ostrik, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
(ostrik@math.mit.edu)
Received June 2, 2001; accepted in revised form August 20, 2001.