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 functions We introduce the (purely algebraic) notion of ann-Hopf algebra and show that the ring of functions on ann-valued group and, in the topological case, the cohomology has ann-Hopf algebra structure. LetRo andR1 be two Kempf-Ness sets arising from moment maps induced by strictly plurisubharmonic,K-invariant, proper functions. A theorem of Richardson states that the algebra of regular functions ofG is a free module over the subalgebra of regular class functions. On correlation functions of drinfeld currents and shuffle algebras We express the vanishing conditions satisfied by the correlation functions of Drinfeld currents of quantum affine algebras, imposed by the quantum Serre relations. The F-valued points of the algebra ofstrongly regular functions of a Kac-Moody group Quantum Symmetric Pairs and Their Zonal Spherical Functions We study the space of biinvariants and zonal spherical functions associated to quantum symmetric pairs in the maximally split case. As a consequence, there is either a unique set, or an (almost) unique two-parameter set of Weyl group invariant quantum zonal spherical functions associated to an irreducible symmetric pair. We compute the ring of ${\mbox{\rm SL}}(2,{\mbox{\bf R}})$-invariants in the ring of polynomial functions, ${\mathcal P}$, on ${\mathcal A}$. Correlation Functions of Strict Partitions and Twisted Fock Spaces Using twisted Fock spaces, we formulate and study two twisted versions of the n-point correlation functions of Bloch-Okounkov, and then identify them with q-expectation values of certain functions on the set of (odd) strict partitions. We find closed formulas for the 1-point functions in both cases in terms of Jacobi θ-functions. These correlation functions afford several distinct interpretations. The Bloch-Okounkov correlation functions at higher levels We establish an explicit formula for the n-point correlation functions in the sense Using these generating sets, we shall determine the Hilbert series of the above Freudenburg's and Daigle and Freudenburg's nonfinitely generated Ga-invariant rings, and find that these Hilbert series are rational functions. Then we also show that the Hilbert series of nonfinitely generated invariant rings appearing in the author's linear counterexamples are rational functions. These eigenfunctions are nonsymmetric versions of the Wilson polynomials and the Wilson functions. Given an automorphism Φ, we denote by k(X)Φ its field of invariants, i.e., the set of rational functions f on X such that f o Φ = f.

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