We study the local $L$ -functions for Levi subgroups in split spinor groups defined via the Langlands-Shahidi method and prove a conjecture on their holomorphy in a half plane. These results have been used in the work of Kim and Shahidi on the functorial product for $\text{G}{{\text{L}}_{2}}\,\times \,\text{G}{{\text{L}}_{3}}$ .

We investigate representations of the Cuntz algebra ${{\mathcal{O}}_{2}}$ on antisymmetric Fock space ${{F}_{a}}({{\mathcal{K}}_{1}})$ defined by isometric implementers of certain quasi-free endomorphisms of the CAR algebra in pure quasi-free states $\varphi {{P}_{1}}$ . We pay special attention to the vector states on ${{\mathcal{O}}_{2}}$ corresponding to these representations and the Fock vacuum, for which we obtain explicit formulae. Restricting these states to the gauge-invariant subalgebra ${{\mathcal{F}}_{2}}$ , we find that for natural choices of implementers, they are again pure quasi-free and are, in fact, essentially the states $\varphi {{P}_{1}}$ . We proceed to consider the case for an arbitrary pair of implementers, and deduce that these Cuntz algebra representations are irreducible, as are their restrictions to ${{\mathcal{F}}_{2}}$ .

The endomorphisms of $B\left( {{F}_{a}}({{\mathcal{K}}_{1}}) \right)$ associated with these representations of ${{\mathcal{O}}_{2}}$ are also considered.

We develop a method for deriving integral representations of certain orthogonal polynomials as moments. These moment representations are applied to find linear and multilinear generating functions for $q$ -orthogonal polynomials. As a byproduct we establish new transformation formulas for combinations of basic hypergeometric functions, including a new representation of the $q$ -exponential function $\text{ }{{\varepsilon }_{q}}$ .

Let $\mathbf{A}$ be a $k$ -element algebra whose chief factor size is $c$ . We show that if $\mathbf{B}$ is in the variety generated by $\mathbf{A}$ , then any abelian chief factor of $\mathbf{B}$ that is not strongly abelian has size at most ${{c}^{k-1}}$ . This solves Problem 5 of The Structure of Finite Algebras, by D. Hobby and R. McKenzie. We refine this bound to $c$ in the situation where the variety generated by $\mathbf{A}$ omits type 1. As a generalization, we bound the size of multitraces of types 1, 2, and 3 by extending coordinatization theory. Finally, we exhibit some examples of bad behavior, even in varieties satisfying a congruence identity.

We introduce the notion of strongly projective graph, and characterise these graphs in terms of their neighbourhood poset. We describe certain exponential graphs associated to complete graphs and odd cycles. We extend and generalise a result of Greenwell and Lovász [6]: if a connected graph $G$ does not admit a homomorphism to $K$ , where $K$ is an odd cycle or a complete graph on at least 3 vertices, then the graph $G\,\times \,{{K}^{S}}$ admits, up to automorphisms of $K$ , exactly $s$ homomorphisms to $K$ .

We study the moderate growth generalized Whittaker functions, associated to a unitary character $\psi $ of a unipotent subgroup, for the non-tempered cohomological representation of $G\,=\,\text{Sp}\left( 2,\,\mathbb{R} \right)$ . Through an explicit calculation of a holonomic system which characterizes these functions we observe that their existence is determined by the including relation between the real nilpotent coadjoint $G$ -orbit of $\psi $ in $\mathfrak{g}_{\mathbb{R}}^{*}$ and the asymptotic support of the cohomological representation.

Willis's structure theory of totally disconnected locally compact groups is investigated in the context of permutation actions. This leads to new interpretations of the basic concepts in the theory and also to new proofs of the fundamental theorems and to several new results. The treatment of Willis's theory is self-contained and full proofs are given of all the fundamental results.

Let $\pi $ be an irreducible generalized principal series representation of $G\,=\,\text{Sp}\left( 2,\,\mathbb{R} \right)$ induced from its Jacobi parabolic subgroup. We show that the space of algebraic intertwining operators from $\pi $ to the representation induced from an irreducible admissible representation of $\text{SL}\left( 2,\,\mathbb{C} \right)$ in $G$ is at most one dimensional. Spherical functions in the title are the images of $K$ -finite vectors by this intertwining operator. We obtain an integral expression of Mellin-Barnes type for the radial part of our spherical function.