Confirming a conjecture of Mark Shimozono, we identify polynomial representatives for the Schubert classes of the affine Grassmannian as the k-Schur functions in homology and affine Schur functions in cohomology. Our results rely on Kostant and Kumar’s nilHecke ring, work of Peterson on the homology of based loops on a compact group, and earlier work of ours on non-commutative k-Schur functions.

Little [Adv. Math. 174 (2003), 236–253] developed a combinatorial algorithm to study the Schur-positivity of Stanley symmetric functions and the Lascoux–Schützenberger tree. We generalize this algorithm to affine Stanley symmetric functions, which were introduced recently in [T. Lam: “Affine Stanley symmetric functions,” Amer. J. Math., to appear].

We study t-analogs of string functions for integrable highest weight representations of the affine Kac-Moody algebra A (1) 1 . We obtain closed form formulas for certain t-string functions of levels 2 and 4. As corollaries, we obtain explicit identities for the corresponding affine Hall-Littlewood functions, as well as higher-level generalizations of Cherednik’s Macdonald and Macdonald-Mehta co...

We proved affine planes corresponding to quadratic planar functions over Fpn are semifield planes, and we determined affine planes corresponding to planar functions f(x) = x10 − αx6 − α2x2 by Ding and Yuan. Moreover we calculated explicit shapes of planar functions from the square mappings of almost all known finite commutative semifields.

Let G be a simple algebraic group of type A or D defined over C and T be a maximal torus of G. For a dominant coweight λ of G, the T -fixed point subscheme (Gr λ G) T of the Schubert variety Gr λ G in the affine Grassmannian GrG is a finite scheme. We prove that there is a natural isomorphism between the dual of the level one affine Demazure module corresponding to λ and the ring of functions (...

Given a hyperplane arrangement A in a real vector space V , we introduce a real algebraic prevariety Z(A), and exhibit the complement ofA in the complexification of V as the total space of an affine bundle over Z(A) with fibers modeled on the dual vector space V . 1 Statement Let V be a real vector space of dimension d, and let f1, . . . , fn be a collection of nonconstant affine linear functio...

We prove a Beurling-Helson type theorem on modulation spaces. More precisely, we show that the only C changes of variables that leave invariant the modulation spaces M(R) are affine functions on R. A special case of our result involving the Sjöstrand algebra was considered earlier by A. Boulkhemair.

We establish the linear independence of time-frequency translates for functions f on R having one sided decay limx∈H, |x|→∞ |f(x)|ec|x| log |x| = 0 for all c > 0, which do not vanish on an affine half-space H ⊂ R.

We analyse the Gauss-Manin system of differential equations— and its Fourier transform—attached to regular functions satisfying a tameness assupmption on a smooth affine variety over C (e.g. tame polynomials on Cn+1). We give a solution to the Birkhoff problem and prove Hodge-type results analogous to those existing for germs of isolated hypersurface singularities.

This paper introduces q-series for a curve singularity (C, 0) in the affine space (Cn, 0) via subspace arrangements. These q-series are certain multivariable generating functions whose coefficients are the characteristic polynomials of subspace arrangements associated with the singularity (C, 0) at various orders; the q is the variable in the characteristic polynomials of the subspace arrangeme...