1.2. The length on Ŵ 9 1.3. Reduction modulo W 9 1.4. More notations 11 1.5. Main definition 12 2. Polynomial representation 13 2.1. Macdonald polynomials 14 2.2. Symmetric polynomials 15 2.3. Using intertwiners 16 2.4. Spherical polynomials 18 2.5. The limit t → 0 19 3. Spherical and Whittaker functions 20 3.1. Gauss-type integrals 21 3.2. Global spherical function 22 3.3. Global Whittaker fun...

Let G be a connected reductive algebraic group defined over a non-Archimedean local field F . The study of Whittaker functions on G has a long history. For example, if G = GL(2), an explicit formula for the Whittaker function is well-known (See [5]). Shintani proved an explicit formula for the case G = GL(n) and expressed it by Schur polynomials in [16]. Casselman and Shalika generalized Shinta...

We introduce and study a one-parameter generalization of the q –Whittaker symmetric functions. This is family multivariate polynomials, whose construction may be viewed as an application procedure fusion from integrable lattice models to vertex model interpretation Hall–Littlewood polynomials [3] , [6] [7] . prove branching Pieri rules, standard dual (skew) Cauchy summation identities, integral...

We study Whittaker functions on nonlinear coverings of simple algebraic groups over a non-archimedean local field. We produce a recipe for expressing such a Whittaker function as a weighted sum over a crystal graph, and show that in type A, these expressions agree with known formulae for the p-part of Multiple Dirichlet Series.

Algorithms for the explicit symbolic and numeric evaluation of Jacquet’s Whittaker function for the GL(n,R) based generalized upper halfplane for n ≥ 2, and an implementation for symbolic evaluation in the Mathematica package GL(n)pack, are described. This requires a comparison of the different definitions of Whittaker function which have appeared in the literature.

Characteristic arborescent communities of mesophytic or xerophytic species have long been recognized as indicative of forest site quality in the Southern Appalachians, where soil moisture availability is the primary environmental variable affecting productivity. But, a workable quantitative system of site classification based on species composition is not available. We devised a prototype exper...

Technical reclamation and spontaneous succession (passive restoration) are the two main approaches for restoring post-mining sites worldwide. Despite substantial differences between both approaches, little is known regarding how they differ in terms of ecological outcomes. We assessed compared vegetation structure composition one forest that 29 years old, technically reclaimed forests 2 10 old ...