A Latin square has six conjugate squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a symmetry if at least two of conjugates are equal. enumerate with and classify them according to several common notions equivalence. also do similar enumerations under additional hypotheses, such as assuming the is reduced, diagonal, idempotent or unipotent. Our data correcte...

We determine the unitary dual of the geometric graded Hecke algebras with unequal parameters which appear in Lusztig’s classification of unipotent representations for exceptional p-adic groups. The largest such algebra is of type F4. Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent represent...

Let G be a connected reductive algebraic group defined over Fq, where q is a power of a prime p that is good for G. Let F be the Frobenius morphism associated with the Fq-structure on G and set G = G , the fixed point subgroup of F . Let P be an F -stable parabolic subgroup of G and let U be the unipotent radical of P; set P = P and U = U . Let Guni be the set of unipotent elements in G. In thi...

In previous work of Gow, Ohmori, Lusztig and the author, the Schur indices of all unipotent characters of finite groups of Lie type have been explicitly determined except for six cases in groups of type F4, E7 and E8. In this paper, we show that the Schur indices of all cuspidal unipotent characters for type F4 and E8 are 1, assuming that the group is defined over a field of “good” characterist...

Abstract. The unipotent variety of a reductive algebraic group G plays an important role in the representation theory. In this paper, we will consider the closure Ū of the unipotent variety in the De Concini-Procesi compactification Ḡ of a quasisimple, adjoint group G. We will prove that Ū − U is a union of some G-stable pieces introduced by Lusztig in [L4]. This was first conjectured by Luszti...

• Basic estimates • The hierarchy of constant terms Let G be a reductive real Lie group, for example G = GL(n,R). Let A be a maximal R-split torus, in the case of GL(n,R) the diagonal matrices, with connected component of the identity A. Let K be a maximal compact subgroup of G, for GL(n,R) the standard orthogonal group O(n). Let N be the unipotent radical of a minimal parabolic containing A, i...

Let G be a reductive algebraic group, P a parabolic subgroup of G with unipotent radical Pu, and A a closed connected unipotent subgroup of Pu which is normalized by P. We show that P acts on A with nitely many orbits provided A is abelian. This generalizes a well-known niteness result, namely the case when A is central in Pu. We also obtain an analogous result for the adjoint action of P on in...

The computations that are suggested by String Theory in the B model requires the existence of degenerations of CY manifolds with maximum unipotent monodromy. In String Theory such a point in the moduli space is called a large radius limit (or large complex structure limit). In this paper we are going to construct one parameter families of n dimensional Calabi-Yau manifolds, which are complete i...

In any semigroup S, we say that elements a and b are left inverses of each other if a = aba, b = bab and a L b, in which case we write a γ b. Right inverses are defined dually with the notation δ. Set τ = δγ. We study the classes of semigroups in which τ has some of the usual properties of a relation. We also consider properties of (maximal) completely simple subsemigroups of S. In terms of the...