Abstract In this paper, we introduce tree varieties as a natural generalization of products of partial flag varieties. We study orbits of the $${{\,\mathrm{\mathbb {P}GL}\,}}$$ action on tree varieties. We characterize tree varieties with finitely many $${{\,\mathrm{\mathbb {P}GL}\,}}$$ orbits, generalizing a celebrated theorem of Magyar, Weyman and Zelevinsky. We give criteria that guarantee that a tree variety has a dense $${{\,\mathrm{\mathbb {P}GL}\,}}$$ orbit and provide many examples of tree varieties that do not have dense $${{\,\mathrm{\mathbb {P}GL}\,}}$$ orbits. We show that a triple of two-step flag varieties $$F(k_1, k_2; n)^3$$ has a dense $${{\,\mathrm{\mathbb {P}GL}\,}}(n)$$ orbit if and only if $$k_1 + k_2 \not = n$$ .

Abstract We study the algebra of functions on the Iwahori group via the category of graded bounded representations of the Iwahori Lie algebra. In particular, we identify the standard and costandard objects in this category with certain generalized Weyl modules. Using this identification we express the characters of the standard and costandard objects in terms of specialized nonsymmetric Macdonald polynomials. We also prove that our category of interest admits a generalized highest weight structure (known as stratified structure). We show, more generally, that such a structure on a category of representations of a Lie algebra implies the Peter-Weyl type theorem for the corresponding algebraic group. In the Iwahori case, standard filtrations of indecomposable projective objects correspond to new “reciprocal” Macdonald-type identities.

Abstract Let $$\mathfrak {g}$$ g be a complex semisimple Lie algebra and let $$\textbf{U}_q(\mathfrak g)$$ U q ( g ) denote the associated Drinfel’d Jimbo quantized enveloping algebra. In this paper we study spherical functions of $$\textbf{U}_q(\mathfrak g)$$ U q ( g ) related to characters. We show invariance under the Wang-Zhang braid group operators and show relative Weyl group invariance, when restricted to the quantum torus.

Abstract Let X be an affine toric variety and let D(X) be the set of weights of all root subgroups. It is known that D(X) together with its embedding into the character group determines X as a toric variety. In this article we prove that X is already determined by the abstract set D(X) together with some additional combinatorial data.