Characters of Nonlinear Groups
Jeffrey Adams
Conference on Representation
Theory of Real Reductive Groups
Salt Lake City, July 30, 2009
slides: www.liegroups.org/talks
www.math.utah.edu/realgroups/conference
Nonlinear Groups
Non Nonlinear Groups
Atlas (lectures last week):
G = connected, complex, reductive, algebraic group
G = G(R)
G L(n, R), S O( p, q), Sp(2n, R) not f
S p(2n, R)
Primary reason for this restriction: Vogan Duality
Atlas parameters for representations of real forms of G:
Z⊂
Y
K i \G/B ×
k
Y
K ∨j \G ∨ /B ∨
j
Vogan duality: Z ∋ (x, y) → (y, x)
Not known in general for nonlinear groups
Outline
Character/representation theory of:
g
(1) G
L(2) (Flicker)
g
(2) G
L(n, Q p ) (Kazhdan/Patterson)
g
(3) G
L(n, R) (A/Huang)
(4) f
S p(2n, R) and S O(2n + 1)
] (G simply laced)
(5) G(R)
Characters and Representations
π = virtual representation of G(R)
P
π = ni=1 ai πi (ai ∈ Z, πi irreducible)
P
θπ = i θπi = virtual character
conjugation invariant function on G(R)0 (regular semisimple
elements)
Identify (virtual) characters and (virtual) representations
(4) f
Sp(2n, R) and S O(2n + 1)
F local, characteristic 0
W symplectic/F, Sp(W ) = Sp(2n, F)
(V, Q): S O(V, Q) = special orthgonal group of (V, Q)
Fix δ ∈ F× /F×2
Proposition [Howe + ǫ] There is a natural bijection
{regular semisimple conjugacy classes in Sp(W )}
and
a
{ (strongly) regular ss conjugacy classes in S O(V, Q)}
(V ,Q)
union: dim(V ) = 2n + 1, det(Q) = δ
Proposition implies relation on characters/representations of Sp(W ),
S O(V, Q)?
Naive guess: π representation of S O(V, Q)
Definition:
Sp(W ))
LiftS O(V ,Q)(θπ )(g) = θπ (g ′ )
(g ↔ g ′ )
= conjugation invariant function on Sp(W )0
Is this the character of a (virtual) representation π ′ of Sp(W )? If so:
f
Sp(2n,R)
Lift S O(V ,Q)(θπ ) = θπ ′
or
f
Sp(2n,R)
Lift S O(V ,Q)(π ) = π ′
Obviously not
Less naive guess:
Sp(W )
Lift S O(V ,Q)(θπ )(g) =
|1 S O (g ′ )|
θπ (g ′ )
|1 Sp (g)|
|1G (g)| = Weyl denominator (absolute value is well defined,
independent of choice of positive roots)
Less obviously not
p:f
S p(W ) → Sp(W ) (metaplectic group)
ψ
ψ
ωψ = ω+ ⊕ ω− = oscillator representation
(choice additive character ψ, see Savin’s lecture. . .
Less naive guess:
Sp(W )
Lift S O(V ,Q)(θπ )(g) =
|1 S O (g ′ )|
θπ (g ′ )
|1 Sp (g)|
|1G (g)| = Weyl denominator (absolute value is well defined,
independent of choice of positive roots)
Less obviously not
p:f
S p(W ) → Sp(W ) (metaplectic group)
ψ
ψ
ωψ = ω+ ⊕ ω− = oscillator representation
(choice additive character ψ, see Savin’s lecture. . . drop it from
notation)
Definition: e
g∈f
Sp(2n, R)0 :
8(e
g) = θω+ (e
g ) − θω− (e
g)
Lemma: e
g∈f
Sp(W )0 , g = p(e
g ) → g ′ ∈ S O(V, Q):
|8(e
g )| =
|1 S O (g ′ )|
|1 Sp (g)|
1
= | det(1 + g)|− 2
Digression: G = Spin(2n), π = spin representation
1
|θπ (e
g)| = | det(1 + g)| 2
Stabilize:
Work only with S O(n + 1, n) (split)
π S O(n + 1, n), θπ is stable if S O(2n + 1, C) conjugation invariant
st
Definition: Sp(2n, R) ∋ g ←→ g ′ ∈ S O(n + 1, n) if g, g ′ have the
same nontrivial eigenvalues
(consistent with [is the stabilization of] earlier definition)
π = stable virtual character of S O(n + 1, n)
Definition:
f
Sp(W )
LiftS O(n+1,n) (θπ )(e
g ) = 8(e
g )θπ (g ′ )
st
( p(e
g ) ←→ g ′ )
Theorem (A, 1998)
f
Sp(W )
Lift S O(n+1,n) is a bijection between
stable virtual representations of SO(n+1,n)
and
stable genuine virtual representations of f
Sp(2n, R)
f
Sp(2n,R)
Write e
π = Lift S O(n+1,n) (π )
f
g ) = θ(e
g ′ ) if
S p(W ): stable means θ(e
(1) p(e
g ) is Sp(2n, C) conjugate to p(e
g′ )
(2) 8(e
g) = 8(e
g ′ ).
proof: Hirai’s matching conditions.
(necessary and sufficient conditions for a function to be the character
of a representation)
Problem: Find an integral transform or other natural realization of this
lifting.
Note: This result (in fact this entire talk) is consistent with, and partly
motivated by, results of Savin (for example his lecture from this
conference)
g
(1)-(3): Lifting from G L(n, F) to G
L(n, F)
(Flicker, Kazhdan-Patterson, A-Huang)
G = G L(n, F) = G L(n) F is p-adic or real
g
p:G
L(n) → G L(n) non-trivial two-fold cover
g
Definition: φ(g) = s(g)2 (s : G L(n) → G
L(n) any section)
g
Definition: h ∈ G L(n), e
g∈G
L(n)
1(h, e
g) =
where τ (h, e
g )2 = 1 . . .
|1(h)|
τ (h, e
g)
|1(e
g )|
g
(1)-(3): Lifting from G L(n, F) to G
L(n, F)
(Flicker, Kazhdan-Patterson, A-Huang)
G = G L(n, F) = G L(n) F is p-adic or real
g
p:G
L(n) → G L(n) non-trivial two-fold cover
g
Definition: φ(g) = s(g)2 (s : G L(n) → G
L(n) any section)
g
Definition: h ∈ G L(n), e
g∈G
L(n)
1(h, e
g) =
|1(h)|
τ (h, e
g)
|1(e
g )|
where τ (h, e
g )2 = 1 . . . (a little tricky to define)
Definition:
g
G L(n)
LiftG
g) = c
L(n) (θπ )(e
X
p(φ(h))= p(e
g)
1(h, e
g )θπ (h)
next result:
Flicker: n = 2, all F
Kazhdan and Patterson: all n, F p-adic
A-Huang: all n, F = R
Theorem: π = virtual representation of G L(n)
g
G L(n)
(1) LiftG
L(n) (θπ ) is (the character of) a virtual representation or 0
g
G L(n)
(2) If π is irreducible and unitary then LiftG
L(n) (θπ ) is ± irreducible
and unitary or 0
g
G L(n)
(3) LiftG
π0 : a small, irreducible, unitary representations
L(n) (C) = e
with infinitesimal character ρ/2 [Huang’s thesis, Wallach’s talk (n=3)]
Remark: Lift commutes with the Euler characteristic of
cohomological induction (surprising)
Remark: Renard and Trapa have an example where π is irreducible
(but not unitary) and Lift(π ) is reducible.
(5) Lifting for simply laced real groups (joint with R. Herb)
G: complex, connected, reductive, simply laced
for this talk assume G d simply connected (ρ exponentiates to G d
suffices)
G(R) real form of G
] → G(R): admissible two-fold cover of G(R)
p : G(R)
(admissible: nonlinear cover of each simple factor for which this
exists)
Recall (Wallach’s talk): nonlinear covers almost always exist
Definition:
φ(g) = s(g)2
] any section)
(g ∈ G(R), s : G(R) → G(R)
Lemma:
(1) φ is well defined (independent of s)
(2) φ induces a map on conjugacy classes
(3) g ∈ H (R) = Cartan ⇒ φ(g) ∈ Z (^
H (R))
proof:
(1) obvious
(2) obvious
(3) obvious
Lemma:
(1) φ is well defined (independent of s)
(2) φ induces a map on conjugacy classes
(3) g ∈ H (R) = Cartan ⇒ φ(g) ∈ Z (^
H (R))
proof:
(1) obvious
(2) obvious
^
(3) obvious (φ(g) ∈ H
(R)0 ⊂ Z (^
H (R)))
[Suppressing for this talk: replace G(R) by G(R) for an (allowed)
quotient G of G - still true, less obvious, need stable in (2)]
]
e
π genuine representation of G(R),
e
g∈^
H (R) = Cartan
Lemma (originally in Flicker)
e
g 6∈ Z (^
H (R)) ⇒ θe
g) = 0
π (e
proof: e
h 6∈ Z (^
H (R))
e
ge
he
g −1 6= e
h
(e
g∈^
H (R))
projecting to H (R) implies
e
ge
he
g −1 = ze
h (p(z)=1)
θπ (e
h) = θπ (e
ge
he
g −1 ) = θπ (ze
h) = −θπ (e
h)
[Heisenberg group over Z/2Z]
Transfer Factors
Assume G is semisimple, simply connected (⇒ G(R) is connected)
H (R) = Cartan, 8+ positive roots
1(g, 8+ ) = eρ (g)
Y
(1 − e−α (g))
α∈8+
Definition: h ∈ H (R)0, e
g∈^
H (R), p(e
g ) = h 2 ∈ H (R) ∩ G(R)0
1(h, e
g) =
1(h, 8+ ) φ(h)
1(g, 8+ ) e
g
p(φ(h)/e
g ) = h 2 / p(e
g ) = 1: φ(h)/e
g = ±1, genuine function in e
g
Obvious: 1(h, e
g ) is independent of choice of 8+ (h ∈ H (R)0 here)
Punt: It is possible to extend the previous construction to general
G(R), and to put conditions on 1(h, e
g ) so that the number of allowed
extensions to H (R) ∩ G(R)0 is acted on simply transitively by
G(R)/G(R)0 .
(hard: reduction to the maximally split Cartan subgroup, Cayley
transforms, need to make the Hirai conditions hold. . . )
So: fix transfer factors 1(h, e
g)
Definition: π = stable virtual representation of G(R):
]
R)
LiftG(
g) = c
G(R) (θπ )(e
X
p(φ(h))= p(e
g)
1(h, e
g )θπ (h)
Theorem: (joint with R. Herb)
]
R)
(1) LiftG(
π
G(R) (θπ ) is the character of a virtual genuine representation e
] or 0
of G(R)
Theorem: (joint with R. Herb)
]
R)
(1) LiftG(
π
G(R) (θπ ) is the character of a virtual genuine representation e
] or 0 - write Lift(π ) = e
of G(R)
π
(2) Infinitesimal character: λ → λ/2
] is a summand of some
(3) Every genuine virtual character of G(R)
]
R)
LiftG(
G(R) (π )
(4) Lift takes (stable) standard modules to (sums of) standard modules
More on (4):
I st (χ) = stabilized standard module defined by character χ of H (R)
P
I st (χ) = w I (wχ) (w ∈ W (M)\Wi )
Lift(I st (χ)) =
X
e
( R)
I (Lift H
H (R) (wχ))
w
proof: Hirai’s matching conditions
Very subtle point: need stability for the matching conditions to hold.
Remark:
(1) Some terms on the RHS are 0.
(2) The non-zero terms on the RHS have distinct central characters.
]
Remark: The notion of stability is probably not interesting for G(R)
in the simply laced case; “L-packets” are (close to) singletons.
Question: Irreducibility of Lift(π )?
]
Remark: The notion of stability is probably not interesting for G(R)
in the simply laced case; “L-packets” are (close to) singletons.
Question: Irreducibility of Lift(π )? Unitarity?
Application to small representations
Related to lectures here by: Savin, Wallach, Kobayashi, Howe;
Application to small representations
Related to lectures here by: Savin, Wallach, Kobayashi, Howe;
Work by Friedberg, Loke, Sanchez, Trapa, Vogan, Weissman, Zhu,
many others. . .
]
R)
Corollary: e
π0 = LiftG(
G(R) (C) is a (non-zero) small virtual genuine
] of infinitesimal character ρ/2.
character of G(R)
Usually (always?) e
π0 is irreducible or the sum of a very small number
of irreducible representations, with distinct central characters
Small: If e
π0 is irreducible, it has Gelfand-Kirillov dimension
≤ 12 (|1| − |1( ρ2 )|)
1( ρ2 ) = {α | h ρ2 , α ∨ i ∈ Z} (integral roots defined by ρ2 )
Character formula I:
(Roughly):
θe
g) =
π0 (e
P
w∈W (1(ρ/2)) sgn(w)e
1(e
g)
(can be made precise)
Direct application of the lifting formula
wρ/2
(e
g)
Character formula II:
e
π0 =
X
±I (^
H (R), χ)
]
(H
(R),e
χ)
where the sum runs over all H (R) and most (all?) genuine irreducible
representations χ of ^
H (R) with dχ = ρ/2
proof: Lift the Zuckerman character formula for C
Other results
Better: replace G(R) with G(R)
f
f
L(2,R)
S L(2,R)
Example: Lift SPG
L(2,R) is better than Lift S L(2,R):
f
R)
Lift SS L(2,
L(2,R)(C) = ω+ + ω+
ψ
ψ
f
L(2,R)
LiftSPG
L(2,R) (C) = ω+
ψ
H (R))
Point: φ(H (R)) is a bigger subset of Z (^
Hard work in A-Herb to allow (certain) G
Two root length case (work in progress with R. Herb); Lifting will be
from real form of G ∨ (R) (generalizing f
Sp(2n, R)/S O(n + 1, n) case)
Vogan Duality for nonlinear groups
Closely related to Lifting, and to the “L-group (?)” for nonlinear
groups.
1) f
Sp(2n, R): D. Renard, P. Trapa
] in type A: Renard, Trapa
2) G(R)
]
3) Spin(2n
+ 1): S. Crofts
] for G simpy laced: A, Trapa
4) G(R)
Long term goal:
Bring nonlinear groups into the Langlands program
or as a first step:
Bring nonlinear groups into the Atlas program