Siddhartha Sahi

Interpolation, integrality, and a generalization of Macdonald's polynomials

In this paper, we introduce a new family of symmetric polynomials which depends on a parameter r. They are defined by specifying certain of their zeros. For the parameter values 1/2, 1, and 2 they have an interpretation in terms of Capelli identities. First, we give explicit formulas in some special cases. Then we show that the polynomials can also be defined in terms of difference equations. As a corollary we obtain that their top homogeneous part is a Jack polynomial. This is used to give a new proof of the Pieri formula for Jack polynomials.In this paper, we introduce a new family of symmetric polynomials which depends on a parameter r. They are defined by specifying certain of their zeros. For the parameter values 1/2, 1, and 2 they have an interpretation in terms of Capelli identities. First, we give explicit formulas in some special cases. Then we show that the polynomials can also be defined in terms of difference equations. As a corollary we obtain that their top homogeneous part is a Jack polynomial. This is used to give a new proof of the Pieri formula for Jack polynomials.

Explicit Hilbert spaces for certain unipotent representations II

LetG be the universal cover of the group of automorphisms of a symmetric tube domain and letP=LN be its Shilov boundary parabolic subgroup. This paper attaches an irreducible unitary representation ofG to each of the (finitely many)L-orbits onn *. The Hilbert space of the representation consists of functions on the orbit which are square-integrable with respect to a certainL-equivariant measure. The representation remains irreducible when restricted toP, and descends to a quotient ofG which is, at worst, thedouble cover of a linear group. If theL-orbit isnot open (inn *), the construction gives a unipotent representation ofG.

Difference equations and symmetric polynomials defined by their zeros

In this paper, we are starting a systematic analysis of a class of symmetric polynomials which, in full generality,was introduced in [Sa]. The main features of these functions are that they are defined by vanishing conditions and that they are nonhomogeneous. They depend on several parameters, but we are studying mainly a certain subfamily which is indexed by one parameter, r. As a special case, we obtain for r = 1 the factorial Schur functions discovered by Biedenharn and Louck [BL]. Our main result is that for general r these functions are eigenvalues of difference operators,which are difference analogues of the Sekiguchi-Debiard differential operators. Thus the functions under investigation are nonhomogeneous variants of Jack polynomials. More precisely, consider the set of partitions of length n, i.e., sequences of integers (λi) with λ1 ≥ · · · ≥ λn ≥ 0. The weight |λ| of a partition λ is the sum of its parts λi. Choose a vector ρ ∈ C which has to satisfy a mild condition. Then, for every λ, there is (up to a constant) a unique symmetric polynomial Pλ of degree at most d which satisfies the following vanishing condition:

The Capelli identity, tube domains, and the generalized Laplace transform☆

The key result in this paper is a proof of a large class of identities, generalizing the Capelli identity [Cl. The Capelli identity is a centerpiece of 19th century invariant theory. It asserts the equality of two differential operators on the n2-dimensional space M(n, R) of n x n real matrices. Let E, denote the ijth elementary matrix, let xii denote the linear functions on M(n, R) dual to E,, and write 3, for the partial differential operator a/ax,. Then this identity says

Kontsevich quantization and invariant distributions on Lie groups

We study Kontsevich’s deformation quantization for the dual of a finite-dimensional real Lie algebra (or superalgebra) g. In this case the Kontsevich ⋆-product defines a new convolution on S(g), regarded as the space of distributions supported at 0 ∈ g. For p ∈ S(g), we show that the convolution operator f 7−→ p ⋆ f is a differential operator with analytic germ. We use this fact to prove a conjecture of Kashiwara and Vergne on invariant distributions on a Lie group G. This yields a new proof of Duflo’s result on local solvability of bi-invariant differential operators on a Lie group G. Moreover, this new proof extends to Lie supergroups.

A new formula for weight multiplicities and characters

We describe a new formula for weight multiplicities and characters of semisimple Lie algebras. Our formula expresses these weight multiplicities as sums of positive rational numbers. In fact, the formula works more generally for the Jacobi polynomials of Heckman and Opdam, which are a one parameter deformation of these characters. As a consequence we prove that after suitable normalization, the coefficients of these polynomials are themselves polynomials in the parameter with positive integral coefficients. This generalizes our earlier joint work with F. Knop on Jack polynomials.

Explicit Hilbert spaces for certain unipotent representations III

Abstract In this paper we construct a family of small unitary representations for real semisimple Lie groups associated with Jordan algebras. These representations are realized on L2-spaces of certain orbits in the Jordan algebra. The representations are spherical and one of our key results is a precise L2-estimate for the Fourier transform of the spherical vector. We also consider the tensor products of these representations and describe their decomposition.

Deformation quantization and invariant distributions

We study Kontsevichs deformation quantization for the dual of a finite-dimensional Lie algebra g. Regarding elements of S(g) as distributions on g, we show that the ★-multiplication operator (r↦r★p) is a differential operator with analytic germ at 0. We use this to establish a conjecture of Kashiwara and Vergne which, in turn, gives a new proof of Duflos result on the local solvability of bi-invariant differential operators on a Lie group.

Degenerate Whittaker functionals for real reductive groups

In this paper we establish a connection between the associated variety of a representation and the existence of certain {\it degenerate} Whittaker functionals, for both smooth and

Generalized and degenerate Whittaker models

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