Square Integrable Representations Midwest Representation Theory - PowerPoint PPT Presentation

Square Integrable Representations Midwest Representation Theory Conference University of Chicago September 5–7, 2014 Joseph A. Wolf University of California at Berkeley – p. 1

Heisenberg Group � w, w ′ � is the standard hermitian inner product on C n h n = i R + C n Heisenberg alg, [ z + w, z ′ + w ′ ] = Im � w, w ′ � H n = i R + C n Heisenberg group: Lie algebra h n H n has center Z = i R , h n has center z = i R Each R –linear functional ξ : C n → R defines a unitary character χ ξ : z + w �→ exp(2 πiξ ( w )) on H n 0 � = λ ∈ z ∗ defines an infinite dimensional irreducible unitary representation π λ of H n with π λ | Z = exp(2 πiλ ) Uniqueness of the Heisenberg commutation relations says that every irreducible unitary representation of H n is equivalent to a χ ξ if it annihilates Z , to a π λ if it does not � z ∗ Θ π λ ( r x f ) | λ | n dλ Fourier inversion has form f ( x ) = c n – p. 2

Kirillov Theory Kirillov used representation theory of H n to give a general theory of unitary reps of csc nilpotent Lie groups N is a csc Lie group, n its Lie algebra, n ∗ dual space of n If f ∈ n ∗ : coadjoint orbit Ad ∗ ( N ) f has invariant symplectic form ω f from b f : n × n → R , b f ( x, y ) = f ([ x, y ]) . polarization : subalgebra p ∈ n s.t. ker b f ⊂ p ⊂ n and p /ker b f maximal null (Lagrangian) subspace of n /ker b f χ f : exp( x ) �→ e 2 πif ( x ) unitary character on P = exp( p ) That defines an (irreducible) unitary rep π f = Ind N P ( χ f ) π f depends (to unitary equiv) only on the orbit Ad ∗ ( N ) f Every irreducible unitary rep of N is equiv to some π f Summary: bijection � N ↔ n ∗ / Ad ∗ ( N ) – p. 3

Heisenberg Group Case H n = i R + C n center Z = i R , h n = i R + C n center z = i R unitary characters χ ξ ( z + w ) = exp(2 πiξ ( w )) for ξ ∈ h ∗ n with ξ | z = 0 (i.e. ξ ( z + w ) = ξ ( w ) ) and Ad ∗ ( N ) ξ = { ξ } . infinite dimensional irreducible unitary representations P (exp(2 πiλ | p )) with 0 � = λ ∈ z ∗ extended to h n by π λ = Ind N λ ( C n ) = 0 . Here Ad ∗ ( N ) λ = { ν ∈ n ∗ | ν | z = λ | z } the coefficients f u,v ( g ) = � u, π λ ( g ) v � of π λ satisfy | f u,v | ∈ L 2 ( N/Z ) . f ( λ ) = trace � the Fourier transform is � N f ( g ) π λ ( g ) dg for f ∈ C ∞ c ( N ) (or even for f ∈ S ( N ) Schwartz space) the Fourier inversion formula is f ( g ) = c � z ∗ � f ( λ ) | λ | n dλ where c depends only on normalization of measures – p. 4

Moore – W. Theory Moore and W. simplified Kirillov theory for csc nilpotent Lie groups with square integrable (modulo center) representations, e.g. Heisenberg group and many others Let N be a csc nilpotent Lie group, n = z + v vector space direct sum where z is its center, n ∗ = z ∗ + v ∗ P : z ∗ → R is the polynomial P ( λ ) = Pf( b λ ) , where Pf( b λ ) is the Pfaffian of the antisymmetric form b λ on n / z The following are equivalent for λ ∈ n ∗ : 1. Ad ∗ ( N ) λ = { ν ∈ n ∗ | ν | z = λ | z } 2. π λ ∈ � N has coefficients in L 2 ( N/Z ) 3. P ( λ ) � = 0 the Fourier inversion formula is f ( g ) = c � z ∗ � f ( λ ) | Pf( λ ) | dλ where c depends only on normalization of measures – p. 5

Upper Triangular Matrices 1 We foliate the upper triangular matrices:     � 0 • • • • • 0 • • • • �   �    0 0 • • • •      � 0 0 • • •     � 0 0 0 • • •     � 0 0 0 • •     or 0 0 0 0 • • •     0 0 0 0 • •     0 0 0 0 0 • •     0 0 0 0 0 •      0 0 0 0 0 0 •  0 0 0 0 0 0 0 0 0 0 0 0 0 Red indicates a normal subgroup L 1 that is a Heisenberg group (the square is its center); blue is a subgroup L 2 that is a Heisenberg group (the square is its center); green is a subgroup L 3 that is a Heisenberg (or abelian) and the square is its center. – p. 6

Upper Triangular Matrices 2   � 0 • • • • •  �  0 • • • •  �  0 0 • • • •   �  0 0  • • •     � 0 0 0   • • •   �   0 0 0 N = or N =  • •      0 0 0 0 • • •     0 0 0 0   • •     0 0 0 0 0   • •    0 0 0 0 0  •     0 0 0 0 0 0   •   0 0 0 0 0 0 0 0 0 0 0 0 0 More generally this gives a decomposition N = L 1 L 2 . . . L m − 1 L m where (a) each L r has unitary reps with coef. in L 2 ( L r /Z r ) , (b) each N r := L 1 L 2 . . . L r is a normal subgp of N with N r = N r − 1 ⋊ L r semidirect product decomposition, (c) Let l r = z r + v r and n = s + v vector space direct sums, s = ⊕ z r , and v = ⊕ v r . Then [ l r , z s ] = 0 and [ l r , l s ] ⊂ v for r > s . – p. 7

Construction of Representations N = L 1 L 2 . . . L m − 1 L m where (a) each L r has unitary reps with coef. in L 2 ( L r /Z r ) , (b) each N r := L 1 L 2 . . . L r is a normal subgp of N with N r = N r − 1 ⋊ L r semidirect, (c) l r = z r + v r , v = ⊕ v r , [ l r , z s ] = 0 and [ l r , l s ] ⊂ v for r > s . 1 with P l 1 ( λ 1 ) � = 0 gives π λ 1 ∈ � λ 1 ∈ z ∗ L 1 2 with P l 2 ( λ 2 ) � = 0 , and π λ 2 ∈ � Then λ 2 ∈ z ∗ L 2 , combines to give π λ 1 + λ 2 ∈ � N 2 with coefficients | f u,v | ∈ L 2 ( N 2 /Z 1 Z 2 ) , || u || 2 || v || 2 In fact || f u,v || 2 L 2 ( N 2 /Z 1 Z 2 ) = | P l 1 ( λ 1 ) P l 2 ( λ 2 ) | . Iterate the construction: λ r ∈ z ∗ r with each P l r ( λ r ) � = 0 , and the square integrable π λ r ∈ � L r , combine to give π λ ∈ � N with coefficients | f u,v | ∈ L 2 ( N/Z 1 ...Z m ) , in fact || u || 2 || v || 2 || f u,v || 2 L 2 ( N/Z 1 ...Z m ) = | P l 1 ( λ 1 ) ...P l m ( λ m ) | . – p. 8

Reformulate S = Z 1 Z 2 . . . Z m has Lie algebra s = z 1 + z 2 + · · · + z m so s ∗ = z ∗ 1 + z ∗ 2 + · · · + z ∗ m λ = λ 1 + λ 2 + · · · + λ m with λ r ∈ z ∗ r view b λ as an antisymmetric bilinear form on n / s P ( λ ) = Pf( b λ ) = P l 1 ( λ 1 ) P l 2 ( λ 2 ) . . . P l m ( λ m ) If P ( λ ) � = 0 then π λ ∈ � N has coefficients | f u,v | ∈ L 2 ( N/S ) L 2 ( N/S = || u || 2 || v || 2 || f u,v || 2 | P ( λ ) | These representations π λ are the stepwise square integrable representations of N . – p. 9

Plancherel Measure & Fourier Inversion π λ has distribution character � f ( g ) π λ ( g ) dg for f ∈ S ( N ) Θ λ ( f ) = trace N Plancherel measure on � N is concentrated on { λ ∈ s ∗ | P ( λ ) � = 0 } and given by ( const ) | P ( λ ) | dλ Fourier inversion formula � s ∗ Θ λ ( r x f ) | P ( λ ) | dλ for f ∈ S ( N ) f ( x ) = ( const ) – p. 10

Compact Quotients N : nilpotent Lie group with stepwise square integrable representations Γ : discrete subgroup with N/ Γ compact in a way that is consistent with the decomposition N = L 1 L 2 . . . L m : Γ ∩ N r cocompact in N r = L 1 L 2 . . . L r for 1 ≦ r ≦ m L 2 ( N/ Γ) = � N mult ( π ) π discrete direct sum with π ∈ � multiplicities mult ( π ) < ∞ mult ( π ) > 0 only for π = π λ with λ integral in the sense that exp(2 πiλ ) is well defined on the torus Z/ (Γ ∩ Z ) Theorem. Let λ ∈ s ∗ with P ( λ ) � = 0 , i.e. with π λ stepwise square integrable. Then (with appropriate normalizations of measures) the multiplicity m ( π λ ) = | P ( λ ) | . – p. 11

Iwasawa Decomposition G real reductive Lie group, G = KAN Iwasawa decomp N maximal unipotent subgroup Theorem. N satisfies the conditions for stepwise square integrable representations N = L 1 L 2 . . . L m − 1 L m where (a) each L r has unitary reps with coef. in L 2 ( L r /Z r ) , (b) each N r := L 1 L 2 . . . L r is a normal subgp of N with N r = N r − 1 ⋊ L r semidirect, (c) l r = z r + v r , v = ⊕ v r , [ l r , z s ] = 0 and [ l r , l s ] ⊂ v for r > s . Idea of proof – at least the construction: { β 1 , . . . , β m } maximal set of strongly orthogonal a –roots (cascade down) ∆ + 1 = { α ∈ ∆ + ( g , a ) | β 1 − α ∈ ∆ + ( g , a ) ∆ + r +1 = { α ∈ ∆ + ( g , a ) \ (∆ + 1 ∪ · · · ∪ ∆ + r ) | β r +1 − α ∈ ∆ + ( g , a ) } l r = g β r + � r g α for 1 ≦ r ≦ m ∆ + Upper triangular matrices: case G = GL ( n ; R ) or SL ( n ; R ) – p. 12

Minimal Parabolics I P = MAN : minimal parabolic subgroup of G , M = Z K ( A ) principal M -orbits on s ∗ : Ad ∗ ( M ) λ where P ( λ ) � = 0 have measurable choice of base points λ b for principal orbits Ad ∗ ( M ) λ with all isotropy subgroups the same a polynomial on s ∗ , defined by Pf , transforms by the modular function of P , and its Fourier transform D is a differential operator on P (or on AN ) that balances lack of unimodularity in the Plancherel formula �� r exp( β r (log a )) dim z r � for a ∈ A , Ad( a )Det s ∗ = Det s ∗ D is an invertible self–adjoint diff op of degree 1 2 (dim n + dim s ) on L 2 ( MAN ) with dense domain C ( MAN ) , and f ( x ) = � P trace π ( D ( r ( x ) f )) dµ P ( π ) � – p. 13