18 problems

Let $G$ be a reductive group over the local field $F$, let $H$ be an endoscopic group, and let $\gamma\in H(F)$ be strongly $G$-regular semisimple. Write $\Lambda_{G,H}(\gamma)$ fo…

Motivic fundamental lemma. For every such pair and every $r\in\mathbb{Q}$, one has the identity in $\mathbb{K}$

The fundamental lemma. The orbital integrals satisfy

Let $G$ be a connected quasi-split reductive group over a non-archimedean local field $F$. Let $\mathcal{SU}_G$ be the vector space of stable distributions on $G(F)$ with true unip…

Let $\alpha:(K_1, K_2) \to \operatorname{Mat}_{2h}(F)$ and $\beta:(K_0, K_3) \to \operatorname{Mat}_{2h}(F)$ be matching regular semisimple pairs. Let $\mathcal{H}_{2h}$ denote the…

Let $F$ be a non-Archimedean local field, let $G$ be the reductive group under consideration, and let $f^{EP}$ be the Euler–Poincaré function with respect to a Haar measure $dg$ on…

Let $K=\mathbb{Q}$, let $\gamma$ be a contributing matrix, and let $S_{\gamma}$ and $D_{\gamma}$ be the positive integer and fundamental discriminant associated with $\gamma$ as in…

Let $F$ be a local non-Archimedean field and let $G$ be the group of $F$-points of a reductive group, with $\hat G_t$ denoting the set of tempered representations. For a regular el…

Let $\mathfrak{s}(E_0)$ be the symmetric Lie algebra and let $\mathfrak{u}(J_1)$ be the Lie algebra of the odd unitary group. Let $Ω(y)$ and $\partial O_y$ denote the transfer fact…

Let $S(E_0)$ be the relevant symmetric space, let $U(J_1)$ be the unitary group attached to the odd hermitian form, and let $Ω(γ)$, $\partial O_γ(1_{S(\mathcal{O}_{E_0})})$, and…

Let $\mathfrak{s}(E_0)$ be the symmetric Lie algebra and let $\mathfrak{u}(J_0)$ and $\mathfrak{u}(J_1)$ be the Lie algebras of the corresponding unitary groups. Let $Ω(y)$ and…

Orbital-integral boundedness conjecture. For any $\gamma\in\mathfrak{g}$,

Assume that $F/F_0$ is ramified and that $n\geq2$ is even. Let $K_0^{{{{{{{{\flat}}}}}}}}$ be the stabilizer of an almost $\pi$-modular lattice and let $K_0^+$ and $K_0^-$ be the s…

Orbital integrals for classical groups are expected to be related only to hyperelliptic curves. The precise formulation below concerns the cohomology of Hessenberg varieties. Hyper…

Let $S(F_0)$ be the symmetric space and let $\mathfrak{s}(F_0)$ be its Lie algebra. Consider the orbital-integral distributions…

Let $F/F_0$ be ramified and let $n\geq3$ be odd. Let $S(F_0)$ be the inhomogeneous symmetric space, let $U_0$ and $U_1$ be the relevant unitary groups, let $K_0\subset U_0(F_0)$ be…

Let $\mathfrak s$ and $\mathfrak s'$ be the tangent spaces of the relevant symmetric spaces, equipped with the smooth-transfer relation defined by the transfer factor $\kappa$, and…

The arithmetic fundamental lemma. For $\gamma\in S_n(F)_{\rm rs}$,