There is a very beautiful correspondence between branched covers of the Riemann sphere P1 and subgroups of the fundamental group π1(P1 − {branch points}), exactly analogous to the correspondence between subfields of an algebraic extension E/F and subgroups of the Galois group Gal(E/F ). This paper explores the concept of a Hecke algebra, which in this context is a generalization of the Galois g...

In [KLM] the authors study certain structure constants for two related rings: the spherical Hecke algebra of a split connected reductive group over a local non-Archimedean field, and the representation ring of the Langlands dual group. The former are defined relative to characteristic functions of double cosets, and the latter relative to highest weight representations. They prove that the nonv...

We begin by recalling standard facts concerning Hecke algebras and modular forms, for details of which the reader is referred to Chapter 3 of Shimura’s book [11]. By definition H (in Shimura’s notation, R(Γ, GL2(Q) )⊗Q) is the Q-algebra spanned by double cosets [ΓγΓ], for γ ∈ GL2(Q) with det γ > 0. Write as usual Mk(Γ) for the complex vector space of holomorphic modular forms of weight k ≥ 0, a...

Let H and K be spherical subgroups of a reductive complex group G. In many cases, detailed knowledge of the double coset space H\G/K is of fundamental importance in group theory and representation theory. If H or K is parabolic, then H\G/K is finite, and we recall the classification of the double cosets in several important cases. If H = K is a symmetric subgroup of G, then the double coset spa...

We find a two-parameter family of solutions the Yang–Mills equations for gauge group SO(1,3) on Minkowski space by foliating different parts it with non-compact coset spaces isometry. The interior lightcone is foliated hyperbolic H3≅SO(1,3)/SO(3), while exterior employs de Sitter dS≅3SO(1,3)/SO(1,2). itself parametrized SO(1,3)/ISO(2) in nilpotent fashion. Equivariant reduction system first two...

We explore classical (relative) difference sets intersected with the cosets of a subgroup of small index. The intersection sizes are governed by quadratic Diophantine equations. Developing the intersections in the subgroup yields an interesting class of group divisible designs. From this and the Bose-Shrikhande-Parker construction, we obtain some new sets of mutually orthogonal latin squares. W...