One-dimensional nil-DAHA and Whittaker functions II

This work is devoted to the theory of nil-DAHA for the root system A₁ and its applications to symmetric and nonsymmetric (spinor) global q-Whittaker functions, integrating the q-Toda eigenvalue problem and its Dunkl-type nonsymmetric version. The spinor global functions extend the symmetric ones to the case of all Demazure characters (not only those for dominant weights); the corresponding Gromov–Witten theory is not known. The main result of the paper is a complete algebraic theory of these functions in terms of induced modules of the core subalgebra of nil-DAHA. It is the first instance of the DAHA theory of canonical-crystal bases, quite non-trivial even for A₁. As the second part of this work, this paper is mainly devoted to the theory of the core subalgebra of nil-DAHA, its induced modules and their applications to the nonsymmetric global Whittaker functions. The first part was about the analytic aspects of our construction and a general algebraic theory of nil-DAHA for A₁.     

Geometric Satake, Springer correspondence and small representations

For a simply-connected simple algebraic group $G$ over $\mathbb C $ , we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of $G$ , generalizing a well-known fact about $GL_n$ . Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group.     

Spherical and Whittaker functions via DAHA I

This paper begins with an exposition of the classical $p$ -adic theory of the Macdonald, Matsumoto, and Whittaker functions aimed at the affine generalizations. The major directions are as follows: (1) extending the theory of DAHA to arbitrary levels and (2) the affine Satake map and Hall functions via DAHA. The key result is the proportionality of the two different formulas for the affine symmetrizer, the Satake-type formula and that based on the polynomial representation of DAHA. The latter approach results in two important formulas for the affine symmetrizer generalizing the relations between the Kac–Moody characters and Demazure characters.     

Spherical and Whittaker functions via DAHA II

This paper is focused on the spinor (nonsymmetric) Whittaker functions in the rank one, related q-Toda–Dunkl operators, and other aspects of the spinor construction, including one-dimensional Bessel functions and the isomorphism between the affine Knizhnik–Zamolodchikov equation and the Quantum many-body problem (the Heckman–Opdam system).     

Green functions of the spectral ball and symmetrized polydisk

The Green function of the spectral ball is constant over the isospectral varieties, is never less than the pullback of its counterpart on the symmetrized polydisk, and is equal to it in the generic case where the pole is a cyclic (non-derogatory) matrix. When the pole matrix is derogatory, the inequality is always strict, and the difference between the two functions depends on the multiplicity of the eigenvalues as roots of the minimal polynomial of that matrix. In particular, the Green function of the spectral ball is not symmetric in its arguments. Additionally, some estimates are given for invariant functions in the symmetrized polydisc, e.g. (infinitesimal versions of) the Carathéodory distance and the Green function, that show that they are distinct in dimension greater or equal to 3.     

Crystal bases of modified quantized enveloping algebras and a double RSK correspondence

We give a new combinatorial realization of the crystal base of the modified quantized enveloping algebras of type A+∞ or A_∞. It is obtained by describing the decomposition of the tensor product of a highest weight crystal and a lowest weight crystal into extremal weight crystals, and taking its limit using a tableaux model of extremal weight crystals. This realization induces in a purely combinatorial way a bicrystal structure of the crystal base of the modified quantized enveloping algebras and hence its Peter-Weyl type decomposition generalizing the classical RSK correspondence.     

A class of identities relating Whittaker and Bessel functions

Identities between Whittaker and modified Bessel functions are derived for particular complex orders. Certain polynomials appear in such identities, which satisfy a fourth order differential equation (not of hypergeometric type), and they themselves can be expressed as particular linear combinations of products of modified Bessel and confluent hypergeometric functions.     

CP methods and the evaluation of negative energy Coulomb Whittaker functions

The CP (constant reference potential perturbation) methods are specialized methods for the solution of Schrödinger equations. They allow big step sizes and are well suited for oscillatory problems. As an illustration of the power of these methods, oscillating Whittaker functions of the second kind are obtained with high accuracy, using a high-order CP-method.     

A path-transformation for random walks and the Robinson-Schensted correspondence

The author and Marc Yor recently introduced a path-transformation with the property that, for belonging to a certain class of random walks on , the transformed walk has the same law as the original walk conditioned never to exit the Weyl chamber . In this paper, we show that is closely related to the Robinson-Schensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of and . The corresponding results for the Brownian model are recovered by Donsker's theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the path-transformation and the Robinson-Schensted algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the Robinson-Schensted algorithm and, moreover, extends easily to a continuous setting.

     

Counting measures,monotone random set functions

Summary This paper arose from work on random processes whose values are measures or more general set functions. Secs. 1–3, which have nothing specifically random, discuss two topologies for certain sigma-finite measures. One, applicable only to counting measures, is a quotient topology which is useful in the finite case but excessively weak in the infinite case. Making use of a well-known result of P. Hall on sets of representatives, we describe this topology and show that it can be enlarged to the stronger one generated by a modification of the Lévy-Prohorov (L-P) metric. Sec. 4 gives a property of the L-P metric for finite integer valued counting measures. The rest of the paper deals with a random monotone non-negative set function in a separable metric space X. If X is complete and if is subadditive and right continuous¹ in probability on certain classes of sets, we show the existence of a version of with right-continuous sample functions. If X is locally compact and is left continuous in probability on a certain class of open sets, there is a left-continuous version. With appropriate additional assumptions, we obtain versions that are measures or capacities. In the latter case, a 0–1 valued set function represents a random closed or compact set. The form of integer-valued strongly subadditive set functions is described for certain cases.Supported in part by National Science Foundation Grant GP-6216     

Whittaker functions for generalized principal series representations of GSp(2,R)

We study Whittaker functions for generalized principal series representations of $\mathrm{GSp}(2,\mathbf{R})$ induced from Siegel parabolic subgroup. We give Mellin–Barnes type integral representations of Whittaker functions belonging to certain K-types. As an application of our explicit formulas, we compute the archimedean parts of Novodvorsky's zeta integrals. As a consequence we can show the entireness of the spinor L-functions for generic cusp forms on $\mathrm{GSp}(2)$.     

On a correspondence between probabilistic and fuzzy choice functions

Probabilistic and fuzzy choice functions are used to describe decision situations in which some degree of uncertainty or imprecision is involved. We propose a way to equate these two formalisms by means of residual implication operations. Furthermore, a set of new rationality conditions for probabilistic choice functions is proposed and proved to be sufficient to ensure that the associated fuzzy choice function is rational.     

Estimates of derivatives of random functions, II

Approximate formulas for the derivatives of functions of several variables contaminated by noise are given. Error estimates are derived. The formulas are optimal in some sense.     

Upper functions for positive random functionals. I. General setting and Gaussian random functions

In this paper we are interested in finding upper functions for a collection of real-valued random variables {Ψ(χ _θ ), θ ∈ Θ}. Here {χ _θ , θ ∈ Θ} is a family of continuous random mappings, Ψ is a given sub-additive positive functional and Θ is a totally bounded subset of a metric space. We seek a nonrandom function U: Θ → ?₊ such that sup _θ∈Θ{Ψ(χ _θ ) ? U(θ)}₊ is “small” with prescribed probability. We apply the results obtained in the general setting to the variety of problems related to Gaussian random functions and empirical processes.     

Extending simulation uses of antithetic variables: partially monotone functions, random permutations, and random subsets

We show how to effectively use antithetic variables to evaluate the expected value of (a) functions of independent random variables, when the functions are monotonic in only some of their variables, (b) Schur functions of random permutations, and (c) monotone functions of random subsets.