Irreducible Representations of Sp(C 2l, Ω) on C 2l Department of Mathematics University of Texas at Arlington Arlington, TX USA March, 2016
We fix G GL C (N) to be a reductive linear algebraic group. By [G] we denote the set of equivalence classes of irreducible representations of G. On the other hand, Ĝ will denote the subset of [G] of equivalence classes of finite-dimensional irreducible representations of G. The corresponding sets of equivalence classes of representations of an associative algebra A will be denoted by [A] or Â.
Remark We write ρ λ : G End(F λ ) for a representative of the class λ, for each λ [G] and denote this representative by (ρ λ, F λ ). By A(G) (or, by C[G]) we denote the group algebra associated with the group G. Remark Every G-module is considered as an A(G)-module and vice-versa.
Example Fix {e i } to be a basis of V = C 2l. Then define {ϕ i } to be a basis of V such that Ω(e i, ϕ j ) = δ ij, where Ω is a non-degenerate skew-symmetric bilinear form. On V, define the exterior product ε : k C 2l k+1 C 2l and the interior product ι : k C 2l k 1 C 2l. Remark Then we have the following relations: {ε(x), ε(y)} = 0, {ι(x ), ι(y )} = 0, {ε(x), ι(x )} = Ω(x, x)id k C 2l.
Example Continued Let E = 2l i=1 ε(e i)ι(ϕ i ) denote the skew-symmetric Euler operator on V. Remark For u k V, Eu = ku. Let Y = ε( 1 2 Id), X = Y, and H = lid E. Remark [E, X ] = 2X, [E, Y ] = 2Y, [Y, X ] = E lid
Recall that for any vector space V, End(V ) is an associative algebra with unity I V, the identity map on V. For any subset U End(V ), let Comm(U) := {T End(V ) TS = ST for any S U} denote the commutant of U. Remark The set Comm(U) forms an associative algebra with unity I V.
Example Continued Theorem On C 2l, Comm(Sp(C 2l )) = Span C {X, H, Y } = sl C (2). A k-vector u k C 2l is called Ω-harmonic when Xu = 0. The k-homogeneous space of Ω-harmonic elements is denoted by H( k C 2l ) = {u k C 2l Xu = 0}. The space of Ω-harmonic is denoted by H( C 2l, Ω).
Let R be a subalgebra of End(W ) such that 1 R acts irreducibly on W. 2 If g G and T R, then (g, T ) ρ(g)t ρ(g 1 ) R defines an action of G on R. Then we denote by R G = {T R ρ(g)t = T ρ(g) for all g G} the commutant of ρ(g) in R.
Remark Since elements of R G commute with elements from A(G), we may define a R G A(G)-module structure on W. Alternatively, we may consider W as a (R G, A(G))-bimodule. Let E λ = Hom G (F λ, W ) for λ Ĝ. Remark Then E λ is an R G -module satisfying Tu(π λ (g)v) = T ρ(g)u(v) = ρ(g)(tu(v)), where u E λ, v F λ, T R G, and g G.
Theorem As an R G A(G)-bimodule, the space W decomposes as W = λ Ĝ E λ F λ. (1) In the above theorem E F stands for the outer (external) tensor product of the R G -module E and of the A(G)-module F.
Example Continued Let F (l k) denote the irreducible representation of sl C (2) with dimension l k + 1. Theorem Then, there exists a canonical decomposition of C 2l as a (sl C (2), Sp(C 2l ))-bimodule, C 2l = l F (l k) H( k C 2l, Ω). k=0
Theorem (Duality) Each multiplicity space E λ is an irreducible R G -module. Further, if λ, µ Ĝ and E λ = E µ as an R G -module, then λ = µ. Theorem (Duality Correspondence) Let σ be the representation of R G on W and let Ĝ denote the set of equivalence classes of the irreducible representation {E λ } of the algebra R G occurring in W. Then the following hold: The representation (σ, W ) is a dirrect sum of irreducible R G -modules, and each irreducible submodule E λ occurs with finite mulitplicity, dim(f λ ). The map F λ E λ is a bijection.
Example Continued Corollary (Duality) As a G-module, V = λ Ĝ F (λ) Hom G (F (λ), ) V. F (λ) Hom G ( F (λ), V ) is an irreducible End G ( V )-module. (ρ, V ) is the direct sum of irreducible End G ( V )-module
Example Continued Corollary k C 2l = [k/2] i=0 Id i H ( k 2i ) C 2l, Ω The space H( j C 2l, Ω) has dimension ( ) ( 2l j 2l j 2), for j = 1,..., l.
Thank you.