In this paper we describe a machinery for homological calculations of representations of FI_G, and use it to develop a local cohomology theory over any commutative Noetherian ring. As an application, we show that the depth introduced by the second author coincides with a more classical invariant from commutative algebra, and obtain upper bounds of a few important invariants of FI_G-modules in terms of torsion degrees of their local cohomology groups.

Motivated by spectral gluing patterns in the Betti Langlands program, we show that for any reductive group G, a parabolic subgroup P, and a topological surface M, the (enhanced) spectral Eisenstein series category of M is the factorization homology over M of the E2-Hecke category H_{G,P} =IndCoh(LS_{G,P}(D^2, S^1)), where LS_{G,P}(D^2, S^1) denotes the moduli stack of G-local systems on a disk together with a P-reduction on the boundary circle. More generally, for any pair of stacks Y → Z satisfying some mild conditions and any map between topological spaces N → M, we define (Y, Z)^{N,M} = Y^N ×_{Z^N} {Z^M} to be the space of maps from M to Z along with a lift to Y of its restriction to N. Using the pair of pants construction, we define an E_n-category H_n(Y, Z) = IndCoh_0((Y, Z)^{S^n−1,D^n})_{Y}), and compute its factorization homology on any d-dimensional manifold M with d ≤ n, \int_M H_n(Y, Z) = IndCoh_0(Y, Z)^{∂(M×D^n−d),M}_{Y^M}, where IndCoh_0 is the sheaf theory introduced by ArinkinGaitsgory and Beraldo. Our result naturally extends previous known computations of Ben-Zvi–Francis–Nadler and Beraldo.

In [2], J. Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets. We continue with our investigation of Fourier coefficients and their implication to the structure of the cuspidal spectrum for symplectic groups ([16] and [20]). As result, we obtain certain characterization and construction of small cuspidal automorphic representations and gain a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, which has impacts to the generalized Ramanujan problem as posted by P. Sarnak in [43].

Let W be an extended affine Weyl group. We prove that the minimal length elements W of any conjugacy class W of W satisfy some nice properties, generalizing results of Geck and Pfeiffer [<i>On the irreducible characters of Hecke algebras</i>, Adv. Math. <b>102</b> (1993), 7994] on finite Weyl groups. We also study a special class of conjugacy classes, the straight conjugacy classes. These conjugacy classes are in a natural bijection with the Frobenius-twisted conjugacy classes of some W -adic group and satisfy additional interesting properties. Furthermore, we discuss some applications to the affine Hecke algebra W

In Berenstein and Rupel (2015), the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category \mathcal{A} to an appropriate q-polynomial algebra. In the case that \mathcal{A} is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in (Ding and Xu, Sci. China Math. 55(10) 2045–2066, 2012). Moreover, if the underlying graph of Q associated with \mathcal{A} is bipartite and the matrix B associated to the quiver Q is of full rank, we show that the image of the algebra homomorphism is in the corresponding quantum cluster algebra.