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].

In this paper, we associate quantum vertex algebras to a certain family of associative algebras $\widetilde{\A}(g)$ which are essentially Ding-Iohara algebras. To do this, we introduce another closely related family of associative algebras $\A(h)$. The associated quantum vertex algebras are based on the vacuum modules for $\A(h)$, whereas $\phi$-coordinated modules for these quantum vertex algebras are associated to $\widetilde{A}(g)$-modules. Furthermore, we classify their irreducible $\phi$-coordinated modules.

Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI_G and VI_q, and answer positively an open question of Nagpal on representation stability theory.

Let V be an N-graded, simple, self-contragredient, C_2-cofinite vertex operator algebra. We show that if the S-transformation of the character of V is a linear combination of characters of V-modules, then the category C of grading-restricted generalized V-modules is a rigid tensor category. We further show, without any assumption on the character of V but assuming that C is rigid, that C is a factorizable finite ribbon category, that is, a not-necessarily-semisimple modular tensor category. As a consequence, we show that if the Zhu algebra of V is semisimple, then C is semisimple and thus V is rational. The proofs of these theorems use techniques and results from tensor categories together with the method of Moore-Seiberg and Huang for deriving identities of two-point genus-one correlation functions associated to V. We give two main applications. First, we prove the conjecture of Kac-Wakimoto and Arakawa that C_2-cofinite affine W-algebras obtained via quantum Drinfeld-Sokolov reduction of admissible-level affine vertex algebras are strongly rational. The proof uses the recent result of Arakawa and van Ekeren that such W-algebras have semisimple (Ramond twisted) Zhu algebras. Second, we use our rigidity results to reduce the "coset rationality problem" to the problem of C2-cofiniteness for the coset. That is, given a vertex operator algebra inclusion U⊗V↪A with A, U strongly rational and U, V a pair of mutual commutant subalgebras in A, we show that V is also strongly rational provided it is C_2-cofinite.

We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.