Characters of Nonlinear Groups Jeffrey Adams Conference on - PowerPoint PPT Presentation

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 ) GL ( n , R ) , SO ( p , q ) , Sp ( 2 n , R ) not � Sp ( 2 n , R ) Primary reason for this restriction: Vogan Duality Atlas parameters for representations of real forms of G : � � K i \ G / B × K ∨ j \ G ∨ / B ∨ Z ⊂ k j Vogan duality: Z ∋ ( x , y ) → ( y , x ) Not known in general for nonlinear groups

Outline Character/representation theory of: (1) � GL ( 2 ) (Flicker) (2) � GL ( n , Q p ) (Kazhdan/Patterson) (3) � GL ( n , R ) (A/Huang) (4) � Sp ( 2 n , R ) and SO ( 2 n + 1 ) (5) � G ( R ) ( G simply laced)

Characters and Representations π = virtual representation of G ( R ) π = � n i = 1 a i π i ( a i ∈ Z , π i irreducible) θ π = � i θ π i = virtual character conjugation invariant function on G ( R ) 0 (regular semisimple elements) Identify (virtual) characters and (virtual) representations

(4) � Sp ( 2 n , R ) and SO ( 2 n + 1 ) F local, characteristic 0 W symplectic/ F , Sp ( W ) = Sp ( 2 n , F ) ( V , Q ) : SO ( 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 � { (strongly) regular ss conjugacy classes in SO ( V , Q ) } ( V , Q ) union: dim ( V ) = 2 n + 1, det ( Q ) = δ

Proposition implies relation on characters/representations of Sp ( W ) , SO ( V , Q ) ? Naive guess: π representation of SO ( V , Q ) Lift Sp ( W )) SO ( V , Q ) (θ π )( g ) = θ π ( g ′ ) ( g ↔ g ′ ) Definition: = conjugation invariant function on Sp ( W ) 0 Is this the character of a (virtual) representation π ′ of Sp ( W ) ? If so: Sp ( 2 n , R ) � Lift SO ( V , Q ) (θ π ) = θ π ′ or � Sp ( 2 n , R ) SO ( V , Q ) (π) = π ′ Lift Obviously not

Less naive guess: SO ( V , Q ) (θ π )( g ) = | � SO ( g ′ ) | Lift Sp ( W ) | � Sp ( g ) | θ π ( g ′ ) | � G ( g ) | = Weyl denominator (absolute value is well defined, independent of choice of positive roots) Less obviously not p : � Sp ( W ) → Sp ( W ) (metaplectic group) ω ψ = ω ψ + ⊕ ω ψ − = oscillator representation (choice additive character ψ , see Savin’s lecture. . .

Less naive guess: SO ( V , Q ) (θ π )( g ) = | � SO ( g ′ ) | Lift Sp ( W ) | � Sp ( g ) | θ π ( g ′ ) | � G ( g ) | = Weyl denominator (absolute value is well defined, independent of choice of positive roots) Less obviously not p : � Sp ( W ) → Sp ( W ) (metaplectic group) ω ψ = ω ψ + ⊕ ω ψ − = oscillator representation (choice additive character ψ , see Savin’s lecture. . . drop it from notation)

g ∈ � Sp ( 2 n , R ) 0 : Definition: � g ) = θ ω + ( � g ) − θ ω − ( � g ) �( � g ) → g ′ ∈ SO ( V , Q ) : g ∈ � Sp ( W ) 0 , g = p ( � Lemma: � g ) | = | � SO ( g ′ ) | | �( � | � Sp ( g ) | = | det ( 1 + g ) | − 1 2 Digression: G = Spin ( 2 n ) , π = spin representation 1 g ) | = | det ( 1 + g ) | | θ π ( � 2

Stabilize: Work only with SO ( n + 1 , n ) (split) π SO ( n + 1 , n ) , θ π is stable if SO ( 2 n + 1 , C ) conjugation invariant st → g ′ ∈ SO ( n + 1 , n ) if g , g ′ have the Definition: Sp ( 2 n , R ) ∋ g ← same nontrivial eigenvalues (consistent with [is the stabilization of] earlier definition) π = stable virtual character of SO ( n + 1 , n ) Definition: st � Sp ( W ) g ) = �( � g )θ π ( g ′ ) ( p ( � g ) → g ′ ) SO ( n + 1 , n ) (θ π )( � Lift ←

Theorem (A, 1998) � Sp ( W ) Lift SO ( n + 1 , n ) is a bijection between stable virtual representations of SO(n+1,n) and stable genuine virtual representations of � Sp ( 2 n , R ) � Sp ( 2 n , R ) Write � π = Lift SO ( n + 1 , n ) (π) � Sp ( W ) : stable means θ( � g ) = θ( � g ′ ) if (1) p ( � g ) is Sp ( 2 n , C ) conjugate to p ( � g ′ ) g ) = �( � g ′ ) . (2) �( �

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)

(1)-(3): Lifting from GL ( n , F ) to � GL ( n , F ) (Flicker, Kazhdan-Patterson, A-Huang) G = GL ( n , F ) = GL ( n ) F is p-adic or real p : � GL ( n ) → GL ( n ) non-trivial two-fold cover Definition: φ( g ) = s ( g ) 2 ( s : GL ( n ) → � GL ( n ) any section) Definition: h ∈ GL ( n ), � g ∈ � GL ( n ) g ) = | �( h ) | �( h , � g ) | τ( h , � g ) | �( � g ) 2 = 1 . . . where τ( h , �

(1)-(3): Lifting from GL ( n , F ) to � GL ( n , F ) (Flicker, Kazhdan-Patterson, A-Huang) G = GL ( n , F ) = GL ( n ) F is p-adic or real p : � GL ( n ) → GL ( n ) non-trivial two-fold cover Definition: φ( g ) = s ( g ) 2 ( s : GL ( n ) → � GL ( n ) any section) Definition: h ∈ GL ( n ), � g ∈ � GL ( n ) g ) = | �( h ) | �( h , � g ) | τ( h , � g ) | �( � g ) 2 = 1 . . . (a little tricky to define) where τ( h , � Definition: � � GL ( n ) g ) = c �( h , � g )θ π ( h ) GL ( n ) (θ π )( � Lift p (φ( h )) = p ( � g )

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 GL ( n ) � GL ( n ) (1) Lift GL ( n ) (θ π ) is (the character of) a virtual representation or 0 � GL ( n ) (2) If π is irreducible and unitary then Lift GL ( n ) (θ π ) is ± irreducible and unitary or 0 � GL ( n ) GL ( n ) ( C ) = � (3) Lift π 0 : a small, irreducible, unitary representations 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 p : � G ( R ) → G ( R ) : admissible two-fold cover of G ( R ) (admissible: nonlinear cover of each simple factor for which this exists) Recall (Wallach’s talk): nonlinear covers almost always exist Definition: ( g ∈ G ( R ), s : G ( R ) → � φ( g ) = s ( g ) 2 G ( R ) any section )

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)]

π genuine representation of � G ( R ) , � g ∈ � H ( R ) = Cartan � Lemma (originally in Flicker) g �∈ Z ( � H ( R )) ⇒ θ � g ) = 0 � π ( � h �∈ Z ( � proof: � H ( R )) g ∈ � g − 1 �= � g � h � h H ( R ) ) � ( � projecting to H ( R ) implies g − 1 = z � g � h � h (p(z)=1) � θ π ( � h ) = θ π ( � g � h � g − 1 ) = θ π ( z � h ) = − θ π ( � h ) [Heisenberg group over Z / 2 Z ]

Transfer Factors Assume G is semisimple, simply connected ( ⇒ G ( R ) is connected) H ( R ) = Cartan, � + positive roots � �( g , � + ) = e ρ ( g ) ( 1 − e − α ( g )) α ∈ � + g ∈ � g ) = h 2 ∈ H ( R ) ∩ G ( R ) 0 Definition: h ∈ H ( R ) 0 , � H ( R ) , p ( � g ) = �( h , � + ) φ( h ) �( h , � �( g , � + ) g � p (φ( h )/ � g ) = h 2 / p ( � g ) = 1: φ( h )/ � g = ± 1, genuine function in � g

g ) is independent of choice of � + ( h ∈ H ( R ) 0 here) Obvious: �( h , � Punt: It is possible to extend the previous construction to general G ( R ) , and to put conditions on �( h , � 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 �( h , � g ) Definition: π = stable virtual representation of G ( R ) : � � G ( R ) g ) = c �( h , � g )θ π ( h ) G ( R ) (θ π )( � Lift p (φ( h )) = p ( � g )

Theorem: (joint with R. Herb) � G ( R ) G ( R ) (θ π ) is the character of a virtual genuine representation � (1) Lift π of � G ( R ) or 0

Theorem: (joint with R. Herb) � G ( R ) G ( R ) (θ π ) is the character of a virtual genuine representation � (1) Lift π of � G ( R ) or 0 - write Lift (π) = � π (2) Infinitesimal character: λ → λ/ 2 (3) Every genuine virtual character of � G ( R ) is a summand of some � G ( R ) G ( R ) (π) Lift (4) Lift takes (stable) standard modules to (sums of) standard modules