Yang's System of Particles and Hecke Algebras

Abstract

SummaryThe graded Hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions. This operator algebra arises from the study of the root system analogue of Yang's system of n particles on the real line with delta function potential. It turns out that the spectral problem for this generalization of Yang's system is related to the problem of finding the spherical tempered representations of the graded Hecke algebra. This observation turns out to be very useful for both these problems. Application of our technique to affine Hecke algebras yields a simple formula for the formal degree of the generic Iwahori spherical discrete series representations.

Spectral transfer morphisms for unipotent affine Hecke algebras

“…This algebra turns out to be the specialization at v of a finite direct sum of mutually isomorphic normalized (in the sense of [54, paragraph 3.1.2]) affine Hecke algebras (called unipotent affine Hecke algebras) defined over L = C[v ±1 ], and has been explicitly determined in all cases [40,51]. The following general result from the theory of types due to [7] (also see [22]) is fundamental to the approach in this paper: (P,δ) 1 ). This defines a natural functorial action of ( θ ) * on the category R(G F u ) uni (by taking tensor products).…”

“…If there would indeed exist a corresponding transfer map, then its transfer map diagram should be obtained by assigning in addition weights to the vertices in K = F (1) m \J , as described in Definition 3.3. These weights turn out to be uniquely determined by the basic property Proposition [54, Proposition 5.2] of spectral transfer maps (applied to the case of residual points), and this also enables us to find these weights w i easily (using the known classification of residual points of [22] and [56]). Our task is then to prove that these eligible diagrams thus obtained are indeed transfer map diagrams.…”

“…, and let denote the nontrivial essentially strict spectral automorphism of E 7 [q]. Then has the following remarkable property (which is easy to check knowing the spectral map diagram): Let λ [3] , λ [111] , λ [21] be the three orbits of residual points of type A 1 × C 3 , and let μ [4] , μ [31] , μ [22] be the three orbits of residual points of type B 4 . Enumerate these as λ i and…”

“…One of these (δ 8 say) is 10-dimensional, and specializes at equal parameters for F 4 to the discrete series [60] with Langlands parameters (F 4 (a3), [4]). The other, δ 8 restricts to the discrete series with Langlands parameters (F 4 (a3), [22]). Comparing with the tables in [64], we see that δ 8 corresponds with (E 7 (a 5 ), − [3]), while δ 8 corresponds to (E 7 (a 5 ), − [21]).…”

“…Let the total number of factors 2 thus obtained be denoted by M. In order to count M, let us write h m ± u ± (x) for the number coordinates of ξ ± (m ± ) which are equal to x (for x ∈ Z ≥0 ) (cf. [22], or [56,Proposition 6.6]…”