Calculating signatures Adams et al. Introduction Character formulas Signatures of Hermitian forms and Hermitian forms unitary representations Char formulas for invt forms Easy Herm KL polys Jeffrey Adams Marc van Leeuwen Peter Trapa Unitarity algorithm David Vogan Wai Ling Yee Representation Theory of Real Reductive Groups, July 28, 2009
Calculating Outline signatures Adams et al. Introduction Introduction Character formulas Hermitian forms Char formulas for Character formulas invt forms Easy Herm KL polys Hermitian forms Unitarity algorithm Character formulas for invariant forms Computing easy Hermitian KL polynomials Unitarity algorithm
Calculating Introduction signatures Adams et al. G ( R ) = real points of complex connected reductive alg G Introduction Problem: find � G ( R ) u = irr unitary reps of G ( R ) . Character formulas Harish-Chandra: � G ( R ) u ⊂ � G ( R ) = quasisimple irr reps. Hermitian forms Char formulas for Unitary reps = quasisimple reps with pos def invt form. invt forms Example: G ( R ) compact ⇒ � G ( R ) u = � G ( R ) = discrete set. Easy Herm KL polys Example: G ( R ) = R ; Unitarity algorithm � � � χ z ( t ) = e zt ( z ∈ C ) G ( R ) = ≃ C � G ( R ) u = { χ i ξ ( ξ ∈ R ) } ≃ i R Suggests: � G ( R ) u = real pts of cplx var � G ( R ) . Almost. . . G ( R ) h = reps with invt form: � � G ( R ) u ⊂ � G ( R ) h ⊂ � G ( R ) . Approximately (Knapp): � G ( R ) = cplx alg var, real pts G ( R ) h ; subset � � G ( R ) u cut out by real algebraic ineqs. Today: conjecture making inequalities computable.
Example: SL ( 2 , R ) spherical reps Calculating signatures Adams et al. G = SL ( 2 , R ) = 2 × 2 real matrices of determinant 1 G acts on upper half plane H � repn E ( ν ) on ν 2 − 1 Introduction Character formulas eigenspace of Laplacian ∆ H . Hermitian forms Spectrum of ∆ H on L 2 ( H ) is ( −∞ , − 1 ] � ν ∈ i R . Char formulas for invt forms Most E ( ν ) irr; always unique irr subrep J ( ν ) ⊂ E ( ν ) . Easy Herm KL Ex: E ( 1 ) = harmonic fns on H ⊃ J ( 1 ) = constant fns polys J ( ν ) ≃ J ( ν ′ ) ⇔ ν = ± ν ′ ⇒ � Unitarity algorithm G sph = { J ( ν ) } ≃ C / ± 1. Cplx conj for real form of � G sph is ν �→ − ν ; real points � G sph , h ≃ ( i R ∪ R ) / ± 1 ⊂ C / ± 1 These are sph Herm reps. Unitary pts (Bargmann): � G sph , u ≃ ( i R ∪ [ − 1 , 1 ]) / ± 1 ⊂ C / ± 1 Moral: have nice families of reps like E ( ν ) ; interesting irreducibles are smaller. . .
Calculating Categories of representations signatures Adams et al. G cplx reductive alg ⊃ G ( R ) real form ⊃ K ( R ) max cpt. Introduction Rep theory of G ( R ) modeled on Verma modules. . . Character formulas Hermitian forms H ⊂ B ⊂ G maximal torus in Borel subgp, Char formulas for h ∗ ↔ highest weight reps invt forms M ( λ ) Verma of hwt λ ∈ h ∗ , Easy Herm KL L ( λ ) irr quot polys Unitarity algorithm Put cplxification of K ( R ) = K ⊂ G , reductive algebraic. ( g , K ) -mod: cplx rep V of g , compatible alg rep of K . Harish-Chandra: irr ( g , K ) -mod � “arb rep of G ( R ) .” X parameter set for irr ( g , K ) -mods I ( x ) std ( g , K ) -mod ↔ x ∈ X J ( x ) irr quot Set X described by Langlands, Knapp-Zuckerman: countable union (subspace of h ∗ )/(subgroup of W ).
Calculating Character formulas signatures Adams et al. Can decompose Verma module into irreducibles � Introduction ( m µ,λ ∈ N ) M ( λ ) = m µ,λ L ( µ ) Character formulas µ ≤ λ Hermitian forms Char formulas for or write a formal character for an irreducible invt forms � ( M µ,λ ∈ Z ) L ( λ ) = M µ,λ M ( µ ) Easy Herm KL polys µ ≤ λ Unitarity algorithm Can decompose standard HC module into irreducibles � ( m y , x ∈ N ) I ( x ) = m y , x J ( y ) y ≤ x or write a formal character for an irreducible � ( M y , x ∈ Z ) J ( x ) = M y , x I ( y ) y ≤ x Matrices m and M upper triang, ones on diag, mutual inverses. Entries are KL polynomials eval at 1.
Calculating Forms and dual spaces signatures V cplx vec space (or alg rep of K , or ( g , K ) -mod). Adams et al. Hermitian dual of V Introduction V h = { ξ : V → C additive | ξ ( zv ) = z ξ ( v ) } Character formulas (If V is K -rep, also require ξ is K -finite.) Hermitian forms Char formulas for invt forms Sesquilinear pairings between V and W Easy Herm KL Sesq ( V , W ) = {� , � : V × W → C , lin in V , conj-lin in W } polys Unitarity algorithm Sesq ( V , W ) ≃ Hom ( V , W h ) , � v , w � T = ( Tv )( w ) . Cplx conj of forms is (conj linear) isom Sesq ( V , W ) ≃ Sesq ( W , V ) . Corr (conj linear) isom is Hermitian transpose Hom ( V , W h ) ≃ Hom ( W , V h ) , ( T h w )( v ) = ( Tv )( w ) . Sesq form � , � T Hermitian if � v , v ′ � T = � v ′ , v � T ⇔ T h = T .
Defining a rep on V h Calculating signatures Suppose V is a ( g , K ) -module. Write π for repn map. Adams et al. Want to construct functor Introduction cplx linear rep ( π, V ) � cplx linear rep ( π h , V h ) Character formulas Hermitian forms using Hermitian transpose map of operators. REQUIRES Char formulas for invt forms twisting by conjugate linear automorphism of g . Easy Herm KL Assume polys σ : G → G antiholom aut, σ ( K ) = K . Unitarity algorithm Define ( g , K ) -module π h ,σ on V h , π h ,σ ( X ) · ξ = [ π ( − σ ( X ))] h · ξ ( X ∈ g , ξ ∈ V h ) . π h ,σ ( k ) · ξ = [ π ( σ ( k ) − 1 )] h · ξ ( k ∈ K , ξ ∈ V h ) . Traditionally use σ 0 = real form with complexified maximal compact K . We need also σ c = compact real form of G preserving K .
Calculating Invariant Hermitian forms signatures Adams et al. V = ( g , K ) -module, σ antihol aut of G preserving K . Introduction A σ -invt sesq form on V is sesq pairing � , � such that Character formulas Hermitian forms � k · v , w � = � v , − σ ( k − 1 ) · w � � X · v , w � = � v , − σ ( X ) · w � , Char formulas for invt forms ( X ∈ g ; k ∈ K ; v , w ∈ V ). Easy Herm KL polys Proposition Unitarity algorithm σ -invt sesq form on V � ( g , K ) -map T : V → V h ,σ : � v , w � T = ( Tv )( w ) . Form is Hermitian iff T h = T. Assume V is irreducible. V ≃ V h ,σ ⇔ ∃ invt sesq form ⇔ ∃ invt Herm form A σ -invt Herm form on V is unique up to real scalar. T → T h � real form of cplx line Hom g , K ( V , V h ,σ ) .
Calculating Invariant forms on standard reps signatures Adams et al. Recall multiplicity formula � Introduction ( m y , x ∈ N ) I ( x ) = m y , x J ( y ) Character formulas y ≤ x Hermitian forms Char formulas for for standard ( g , K ) -mod I ( x ) . invt forms Want parallel formulas for σ -invt Hermitian forms. Easy Herm KL polys Need forms on standard modules. Unitarity algorithm Form on irr J ( x ) d eformation − − − − − − − → Jantzen filt I n ( x ) on std, nondeg forms � , � n on I n / I n + 1 . Details (proved by Beilinson-Bernstein): I ( x ) = I 0 ⊃ I 1 ⊃ I 2 ⊃ · · · , I 0 / I 1 = J ( x ) I n / I n + 1 completely reducible [ J ( y ): I n / I n + 1 ] = coeff of q ( ℓ ( x ) − ℓ ( y ) − n ) / 2 in KL poly Q y , x = � def Hence � , � I ( x ) n � , � n , nondeg form on gr I ( x ) . Restricts to original form on irr J ( x ) .
Calculating Virtual Hermitian forms signatures Adams et al. Z = Groth group of vec spaces . Introduction Character formulas These are mults of irr reps in virtual reps. Hermitian forms Z [ X ] = Groth grp of finite length reps . Char formulas for invt forms For invariant forms. . . Easy Herm KL polys W = Z ⊕ Z = Groth grp of fin diml forms . Unitarity algorithm Ring structure ( p , q )( p ′ , q ′ ) = ( pp ′ + qq ′ , pq ′ + q ′ p ) . Mult of irr-with-forms in virtual-with-forms is in W : W [ X ] ≈ Groth grp of fin lgth reps with invt forms . Two problems: invt form � , � J may not exist for irr J ; and � , � J may not be preferable to −� , � J .
Calculating Hermitian KL polynomials: multiplicities signatures Adams et al. Fix σ -invt Hermitian form � , � J ( x ) on each irr admitting Introduction one; recall Jantzen form � , � n on I ( x ) n / I ( x ) n + 1 . Character formulas Hermitian forms MODULO problem of irrs with no invt form, write � Char formulas for ( I n / I n − 1 , � , � n ) = w y , x ( n )( J ( y ) , � , � J ( y ) ) , invt forms y ≤ x Easy Herm KL polys coeffs w ( n ) = ( p ( n ) , q ( n )) ∈ W ; summand means Unitarity algorithm p ( n )( J ( y ) , � , � J ( y ) ) ⊕ q ( n )( J ( y ) , −� , � J ( y ) ) Define Hermitian KL polynomials � w y , x ( n ) q ( l ( x ) − l ( y ) − n ) / 2 ∈ W [ q ] Q σ y , x = n Eval in W at q = 1 ↔ form � , � I ( x ) on std. Reduction to Z [ q ] by W → Z ↔ KL poly Q y , x .
Calculating Hermitian KL polynomials: characters signatures Adams et al. Introduction Matrix Q σ y , x is upper tri, 1s on diag: INVERTIBLE. Character formulas Hermitian forms def P σ = ( − 1 ) l ( x ) − l ( y ) (( x , y ) entry of inverse ) ∈ W [ q ] . Char formulas for x , y invt forms Easy Herm KL Definition of Q σ x , y says polys � Q σ ( gr I ( x ) , � , � I ( x ) ) = x , y ( 1 )( J ( y ) , � , � J ( y ) ); Unitarity algorithm y ≤ x inverting this gives � ( − 1 ) l ( x ) − l ( y ) P σ ( J ( x ) , � , � J ( x ) ) = x , y ( 1 )( gr I ( y ) , � , � I ( y ) ) y ≤ x Next question: how do you compute P σ x , y ?
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