Unitary representations of the icosahedral graded Hecke algebra

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1 Unitary representations of the icosahedral graded Hecke algebra AMS Southeastern Section Meeting, LA - March 9, 008 Cathy Kriloff Idaho State University krilcath@isu.edu Joint work in progress with Yu Chen Idaho State University Partial support provided by a grant from the ISU WeLEAD Project which is funded by NSF ADVANCE grant SBE Slides will be available at krilcath/pubs. 1

2 (1) Icosahedral group and graded Hecke algebra () Tempered and unitary representations (3) Intertwining operators and other ingredients () Statement of results for H(H 3 )

3 Icosahedral reflection group H 3 = 10 symmetries of the icosahedron or dodecahedron = Coxeter group s 1, s, s 3 with relations given by h R = reflection representation of W containing root system R Positive roots α 1,... α 1, and actions of s i are given below. Figure 1. Positive roots of H 3, actions of s i, and the root order α 1 α s α 3 s 3 s 1 s α α s α 3 s3 s s 1 s 1 α 7 s1 α 8 α 9 s 3 s α 10 α 11 s s 1 s 3 α 1 α 13 s 1 s 3 α 1 s α1 where for τ = 1+, α = α 1 + τα α = τα 1 + α α = α + τα 3 α 7 = τα 1 + τα α 8 = α 1 + τα + τα 3 α 9 = τα 1 + α + α 3 α 10 = τα 1 + τα + τα 3 α 11 = τα 1 + (τ + 1)α + α 3 α 1 = τα 1 + (τ + 1)α + τα 3 α 13 = (τ + 1)α 1 + (τ + 1)α + α 3 α 1 = (τ + 1)α 1 + (τ + 1)α + τα 3 α 1 = (τ + 1)α 1 + τα + τα 3

4 Why noncrystallographic root systems? Crystallographic graded Hecke algebras have been well-studied by geometric, algebraic, and combinatorial means. Noncrystallographic algebras have no known geometry, but the representation theory is similar in many ways and intriguingly different in others. Understanding the representation theory of noncrystallographic graded Hecke algebras may help reveal some as yet unknown geometry. Efforts to generalize aspects of Lie theory to the setting of complex reflection groups must include the noncrystallographic cases. Since these are still real reflection groups they may serve as good boundary or test cases. Since they differ in fundamental ways they may help show what differences need to be accounted for. On a combinatorial level, these or similar algebras may help to relate objects that exhibit common numerology (e.g., noncrossing partitions, nonnesting partitions, vertices in generalized associahedra) but whose connections are not entirely understood.

5 Associated geometric and combinatorial objects Reflecting hyperplanes H 0 α i = {x h R x, α i = 0} for 1 i 1 Affine hyperplanes H 1 α i = {x h R x, α i = 1} for 1 i 1 C = closure of fundamental chamber is divided into 3-, -, 1-, and 0-dimensional cells by {H 1 α i } 1 i= y x 8 8 Characterize the 0-dimensional cells (vertices) using Z(x) = {α i x, α i P (x) = {α i x, α i = 0} = reflecting hyperplanes containing x = 1} = affine hyperplanes containing x. Theorem(Chen-K, 007). Ideals in the poset of positive roots under the root order characterize the 3-dimensional cells. (See final page.)

6 (Icosahedral) graded Hecke algebra H Additional ingredients: W acts on complexification h C = C R h R. W acts on symmetric algebra S(h C ) = polynomials on h C. { } CW = a w t w a w C with multiplication as in W. w W Graded Hecke algebra is H = CW S(h C ) with multiplication as in S(h C ) and in CW and xt si = t si s i (x) + x, αi = t si s i (x) + x s i(x) α i for x h C. Center is Z(H) = S(h C )W = W -invariant polynomials on h C. Z(H) acts on simple H-module V by central character γ h C (W γ). We will often restrict to real central character γ h R. Goal: Describe the real support of spherical unitary representations of H(H 3 ) ( SPUD ).

7 Tempered representations of H Any simple H-module V has generalized weight space decomposition V = γ h C V gen γ, where γ called a weight of V if V gen γ 0. H-module V is tempered (discrete series) if for all weights λ h C of V and all fundamental weights ω i h C, ω i(λ) 0 (ω i (λ) < 0). Unitary representations of H -operation defined on H by t w = t w 1, for w W and x h. x = t w0 (w 0 x)t w0 = x + α R + x(α )t sα, H-module V is Hermitian if there is nondegenerate form such that hv 1, v = v 1, h v, for all h H, v 1, v V, and V is unitary if, is also positive definite. An invariant Hermitian form on V is equivalent to an H-module isomorphism from V to its Hermitian dual.

8 Langlands classification for H Fix subset Π M of simple roots and root system R M they generate. H has subalgebra H M = H M0 S(t ) (h C, R M) where H M0 (C Π M, R M ), and define t = {ν h C α(ν) = 0 for all α Π M } t = {λ h C λ(α ) = 0 for all α Π M } t + = {ν t Re(α(ν)) > 0 for all α Π \ Π M }. For an H M -module U let I(M, U) = H HM U. Theorem(Langlands Classification - Evens, 199). (1) Every simple H-module is a quotient of a standard module X(M, σ, ν) = I(M, σ C ν ), σ a tempered H M0 -module, ν t +. () X(M, σ, ν) has unique irreducible (Langlands) quotient L(M, σ, ν). (3) L(M, σ, ν) is unique up to conjugacy of the triple (M, σ, ν). The W -module structure of X(M, σ, ν) is constant on cells.

9 Intertwining operator Restrict to spherical representations V such that Hom W (V, 1) 0. Writing w 0 = s i1 s il = (s 1 s s 1 s s 3 ) 3 induces the ordering β 1 = s l s α 1,..., β l 1 = s l α l 1, β l = α l. {α 1, α, α 7, α, α 13, α, α 11, α 1, α 1, α 1, α 9, α 10, α 8, α, α 3 }. For each ν V, define the following element of R[W ]: A(ν) = (1 + ν, β 1 s i1 ) (1 + ν, β s i ) (1 + ν, β l s il ) = (1 + ν, α 1 s 1 ) (1 + ν, α s ) (1 + ν, α 3 s 3 ). A(ν) is independent of the choice of reduced expression for w 0. A(ν) is invertible if and only if ν, αi = ±1 for all i. Define the principal series module X(ν) = H S(h C ) C ν for ν h R. If ν C then X(ν) has a unique simple quotient L(ν), which is spherical, and any spherical H-module appears in this way. L(ν) is Hermitian if and only if w 0 ν = ν. A(ν) is an intertwining operator from X(ν) to X( ν) with image L(ν) if w 0 ν = ν. L(ν) is unitary if and only if ρ(a(ν)) is positive semi-definite for all ρ Ŵ. The signature of ρ(a(ν)) is constant on cells.

10 Results for H(H 3 ) The spherical unitary dual of H(H 3 ) consists of three connected closed 3-dimensional pieces connected to three 1-dimensional pieces, a single isolated 1-dimensional piece, and two isolated points.!"α, $% &'( )'*&!" α+!"α # $%&'( )'*&!" α+ γ γ γ!"α, $% &'( )'*&!" α # γ γ γ γ γ γ γ

11 Techniques (1) Obtain vertices ν of 3-d polyhedral regions (a) Convert half-space description from exact inequalities over Q[ ] to approximate inequalities. (b) Use Polymake to convert approximate half-space description to vertex description. (Is there an algorithm over Q[ ]?) (c) Compute inner products in Maple to guess P (ν), Z(ν) for each vertex ν. (d) Solve systems of equations exactly in Maple to obtain an exact form for each vertex over Q[ ]. () Analyze signs of eigenvalues for each ρ(a(ν)) where ρ is an irreducible representation of H 3 (from W -graphs), ν is a vertex or average of vertices (chosen efficiently) (a) Compute matrix ρ(a(ν)) and characteristic polynomial χ ρ,ν over Q[ ]. (b) Analyze signs of coefficients in χ ρ,ν by inspection if possible and by computing an approximation if not. (c) Use Descartes Rule of Signs to detect or rule out negative eigenvalues.

12 Figure. Ideals in the positive root poset or 3-dim cells for H