Torus orbits on homogeneous varieties and Kac polynomials of quivers

In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients

Fourier transform from the symmetric square representation of PGL2PGL_2 and SL2SL_2

Let $G$ be a connected reductive group over $\overline{\mathbb{F}}_q$ and let $\rho^\vee:G^\vee\rightarrow GL_n$ be an algebraic representation of the dual group $G^\vee$. Assuming that $G$ and $\rho^\vee$ are defined over $\mathbb{F}_q$, Braverman and Kazhdan defined an operator on the space $\mathcal{C}(G(\mathbb{F}_q))$ of complex valued functions on $G(\mathbb{F}_q)$. In this paper we are interested in the case where $G$ is either $SL_2$ or $PGL_2$ and $\rho^\vee$ is the symmetric square representation of $G^\vee$. We construct a natural $G\times G$-equivariant embedding $G\hookrightarrow\mathcal{G}=\mathcal{G}_\rho$ and an involutive operator (Fourier transform) $\mathcal{F}^{\mathcal{G}}$ on the space of functions $\mathcal{C}(\mathcal{G}(\mathbb{F}_q))$ that extends Braverman-Kazhdan's operator