Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

This article is the first in a series that describes a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. (For simplicity, we only consider the untwisted and simply connected case here.) The usual geometric Satake isomorphism for a reductive group G identifies the tensor category Rep(G∨) of finite-dimensional representations of the Langlands dual group G∨ with the tensor category PervG(O)(GrG) of G(O)-equivariant perverse sheaves on the affine Grassmannian GrG=G(K)/G(O) of G. (Here K=C((t)) and O=C[[t]].) As a by-product one gets a description of the irreducible G(O)-equivariant intersection cohomology (IC) sheaves of the closures of G(O)-orbits in GrG in terms of q-analogs of the weight multiplicity for finite-dimensional representations of G∨. The purpose of this article is to try to generalize the above results to the case when G is replaced by the corresponding affine Kac-Moody group Gaff. (We refer to the (not yet constructed) affine Grassmannian of Gaff as the double affine Grassmannian.) More precisely, in this article we construct certain varieties that should be thought of as transversal slices to various Gaff(O)-orbits inside the closure of another Gaff(O)-orbit in GrGaff. We present a conjecture that computes the intersection cohomology sheaf of these varieties in terms of the corresponding q-analog of the weight multiplicity for the Langlands dual affine group Gaff∨, and we check this conjecture in a number of cases. Some further constructions (such as convolution of the corresponding perverse sheaves, analog of the Beilinson-Drinfeld Grassmannian, and so forth) will be addressed in another publication

https://publications.hse.ru/mirror/pubs/share/folder/pkj4i2q8k7/direct/63401902

Pursuing the double affine Grassmannian, I: Transversal slices via instantons on $A_k$-singularities

A permutation sequence $(\sigma_n)_{n \in \mathbb{N}}$ is said to be convergent if, for every fixed permutation $\tau$, the density of occurrences of $\tau$ in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function $Z:[0,1]^2 \to [0,1]$ with the additional properties that, for every fixed $x \in [0,1]$, the restriction $Z(x,\cdot)$ is a cumulative distribution function and, for every $y \in [0,1]$, the restriction $Z(\cdot,y)$ satisfies a "mass" condition. This limit process is well-behaved: every function in the class of limit objects is a limit of some permutation sequence, and two of these functions are limits of the same sequence if and only if they are equal almost everywhere. An important ingredient in the proofs is a new model of random permutations, which generalizes previous models and is interesting for its own sake.

Limits of permutation sequences through permutation regularity

This paper is a continuation of a previous paper in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dimensional case.

Iwahori-Hecke algebras for p-adic loop groups

We introduce a generalization of the classical Hall-Littlewood and Kostka-Foulkes polynomials to all symmetrizable Kac-Moody algebras. We prove that these Kostka-Foulkes polynomials coincide with the natural generalization of Lusztig's $t$-analog of weight multiplicities, thereby extending a theorem of Kato. For $g$ an affine Kac-Moody algebra, we define $t$-analogs of string functions and use Cherednik's constant term identities to derive explicit product expressions for them.

Kostka-Foulkes polynomials for symmetrizable Kac-Moody algebras.

Unique factorization of tensor products for Kac–Moody algebras

We show that the rank 10 hyperbolic Kac–Moody algebra E10 contains every simply laced hyperbolic Kac–Moody algebra as a Lie subalgebra. Our method is based on an extension of earlier work of Feingold and Nicolai.

Embeddings of Hyperbolic Kac Moody Algebras into E 10

We give a new interpretation of the chromatic polynomial of a simple graph $$G$$G in terms of the Kac---Moody Lie algebra $$\mathfrak {g}$$g with Dynkin diagram $$G$$G. We show that the chromatic polynomial is essentially the $$q$$q-Kostant partition function of $$\mathfrak {g}$$g evaluated on the sum of the simple roots. Applying the Peterson recurrence formula for root multiplicities of $$\mathfrak {g}$$g, we obtain a new realization of the chromatic polynomial as a weighted sum of paths in the bond lattice of $$G$$G.

/pdf/chromatic-polynomials-of-graphs-from-kac-moody-algebras-2s6h9ln1n5.pdf

Sankaran Viswanath

Papers

Kostka-Foulkes polynomials for symmetrizable Kac-Moody algebras.

Unique factorization of tensor products for Kac–Moody algebras

Embeddings of Hyperbolic Kac Moody Algebras into E 10

Chromatic polynomials of graphs from Kac---Moody algebras

Demazure flags, Chebyshev polynomials, partial and mock theta functions