functions 
We introduce the (purely algebraic) notion of annHopf algebra and show that the ring of functions on annvalued group and, in the topological case, the cohomology has annHopf algebra structure.


LetRo andR1 be two KempfNess sets arising from moment maps induced by strictly plurisubharmonic,Kinvariant, 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 Fvalued points of the algebra ofstrongly regular functions of a KacMoody 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 twoparameter 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 npoint correlation functions of BlochOkounkov, and then identify them with qexpectation values of certain functions on the set of (odd) strict partitions.


We find closed formulas for the 1point functions in both cases in terms of Jacobi θfunctions.


These correlation functions afford several distinct interpretations.


The BlochOkounkov correlation functions at higher levels


We establish an explicit formula for the npoint 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 Gainvariant 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.

