This note constructs the flat toric degeneration of the manifold Fln of flags in Cn from [GL96] as an explicit GIT quotient of the Gröbner degeneration in [KM03]. This implies that Schubert varieties degenerate to reduced unions of toric varieties, associated to faces indexed by rc-graphs (reduced pipe dreams) in the Gelfand–Cetlin polytope. Our explicit description of the toric degeneration of...

For an arbitrary simple Lie algebra g and an arbitrary root of unity q, the closed subsets of the Weyl alcove of the quantum group Uq(g) are classified. Here a closed subset is a set such that if any two weights in the Weyl alcove are in the set, so is any weight in the Weyl alcove which corresponds to an irreducible summand of the tensor product of a pair of representations with highest weight...

We study the Ramond twisted representations of the affine W algebra W(ḡ, f) in the case that f is Richardson. We establish the vanishing and the almost irreducibility of the corresponding BRST cohomology. This confirms some of the recent conjectures of Kac and Wakimoto [KW5]. In type A, our results give the characters of all irreducible ordinary Ramond twisted representations of W(sln, f) for a...

It is shown that the support of an irreducible weight module over the SchrödingerVirasoro Lie algebra with an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite-dimensional. As a side-product, it is obtained that every simple weight module over the Schrödinger-Virasoro Lie algebra with a nontrivial finite-dime...

For a polynomial f(T ) ∈ Z[T ], the frequency with which the values f(n) are prime has been considered since at least the 18-th century. Euler observed, in a letter to Goldbach in 1752, that the sequence n + 1 has many prime values for 1 ≤ n ≤ 1500. Legendre assumed an arithmetic progression an+ b with (a, b) = 1 contains infinitely many primes in his work on the quadratic reciprocity law. Ther...

Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let W = V ⊕ Fe with the form Q extending q with Q(e) = 1. Consider the standard embedding O(V ) →֒ O(W ) and the two-sided action of O(V ) ×O(V ) on O(W ). In this note we show that any O(V )×O(V )-invariant distribution on O(W ) is invariant with respect to transposition. This result was...

In this paper, we consider lifts of π-partial characters with the property that the irreducible constituents of their restrictions to certain normal subgroups are also lifts. We will show that such a lift must be induced from what we call an inductive pair, and every character induced from an inductive pair is a such a lift. With this condition, we will get a lower bound on the number of such l...

Let Φ be an irreducible crystallographic root system with Weyl group W and coroot lattice Q̌, spanning a Euclidean space V . Let m be a positive integer and AmΦ be the arrangement of hyperplanes in V of the form (α, x) = k for α ∈ Φ and k = 0, 1, . . . ,m. It is known that the number N(Φ, m) of bounded dominant regions of AmΦ is equal to the number of facets of the positive part ∆ m + (Φ) of the...

It is well-known that one of the most important ways of constructing cohomology for an arithmetic quotient Γ\X, where X is the symmetric space associated with a semi-simple Lie group G with finite center and Γ is a discrete subgroup of G, is through the use of Matsushima’s formula, namely by exhibiting irreducible unitary representations with non-zero continuous cohomology which occur in L2(Γ\G...

We give quantitive estimates for the number of locally symmetric spaces of a given type with bounded volume. Explicitly, let S be a symmetric space of non-compact type without Euclidean de Rham factors. Then, after rescaling appropriately the Riemannian metric, the following hold. Theorem A If rank(S) = 1 and S ≇ H2,H3, then there are at most V V Riemannian manifolds, locally isometric to S, wi...