We calculate a G_2-period of a Fourier coefficient of a cuspidal Eisenstein series on the split simply-connected group E_6, and relate this period to the Ginzburg-Rallis period of cusp forms on GL_6. This gives us a relation between the Ginzburg-Rallis period and the central value of the exterior cube L-function of GL_6.

Suppose G is a reductive algebraic group, T is a Cartan subgroup of G, N= Norm (T), and W= N/T is the Weyl group. If w W has order d, it is natural to ask about the orders lifts of w to N. It is straightforward to see that the minimal order of a lift of w has order d or 2d, but it can be a subtle question which holds. We first consider the question of when W itself lifts to a subgroup of N (in which case every element of W lifts to an element of N of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all lifts of w are conjugate, and therefore have the same order. We also consider the twisted case.

For each integer we describe the space of stability conditions on the derived category of the n-dimensional Ginzburg algebra associated to the A2 quiver. The form of our results points to a close relationship between these spaces and the Frobenius-Saito structure on the unfolding space of the A2 singularity.

We study the resonances of the Laplacian acting on the compactly supported sections of a homogeneous vector bundle over a Riemannian symmetric space of the non-compact type. The symmetric space is assumed to have rank-one but the irreducible representation τ of K defining the vector bundle is arbitrary. We determine the resonances. Under the additional assumption that τ occurs in the spherical principal series, we determine the resonance representations. They are all irreducible. We find their Langlands parameters, their wave front sets and determine which of them are unitarizable.

Varagnolo and Vasserot conjectured an equivalence between the category O for CRDAHA’s and a subcategory of an affine parabolic category O of type A. We prove this conjecture. As applications, we prove a conjecture of Rouquier on the dimension of simple modules of CRDAHA’s and a conjecture of Chuang–Miyachi on the Koszul duality for the category O of CRDAHA’s.