19 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 $n$, $k$, and $r$ be the parameters indexing the relevant classical groups and equal-valuation strips. Let $\Theta^{G,+}_{n,k,r}$ and $\Theta^{G,-}_{n,k,r}$ be the Chow motives…

Let ${\mathfrak g}=\operatorname{so}(2n+1)$ and let ${\mathfrak h}=\operatorname{so}(2k+1)\times\operatorname{so}(2n-2k+1)$. Let $Y$ and $Z$ be regular semisimple elements, and sup…

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…

Relative fundamental lemma in the semi-Lie case.

Let $D$ be a central simple algebra and let $G$ and $G'$ be the groups in the Guo–Jacquet comparison, with test functions $f_D=1_{O_D^\times}$ and $f'_D$ as defined from the standa…

Let $\tilde{G}=G\rtimes\theta$ be a twisted space with an involution $\theta$ on a reductive group $G$, and let $H$ be an endoscopic group of $\tilde{G}$. Assume that $G$ and $H$ a…

Fundamental lemma. When $p$ is odd, $1_{\mathfrak{k}}$ and $\{1_{\mathfrak{k}_0},0\}$ are transfers.

Let $V$ be a hermitian space over $E$, and let $I(x,j)$ be the number of self-dual lattices satisfying the stated stability conditions. Let $O(x,j)$ be the corresponding signed lat…

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…

Assume the relevant unramified hermitian-space setup and let $K_0(\varpi)$, $K'$, $\mathfrak{k}'$, $K_1^{{{{{{{{\flat}}}}}}}}$, and $K_1$ be the compact subgroups and lattices defi…

Assume that $F/F_0$ is unramified, that the special vectors $u_i$ have norm $\varpi$, and that $K'=S_n(O_{F_0})\cap K_0(\varpi)$, where $K_0(\varpi)$ is the subgroup of matrices co…

Fundamental lemma. It is conjectured that there is a matching relation between a Hecke function on $S(F)$ and a Hecke function on $\widetilde{\mathrm{GL}}_r(F)$.

Assume that $F/F_0$ is unramified. Let $S_n(F_0)$ be the symmetric space and let $W_0$ be the relevant split hermitian space. Choose the natural transfer factor…

BC-conjecture. If $u$ and $v$ are $BC$-matching, then their stable orbital integrals of the unit elements satisfy

Lemme fondamental pondéré non standard. Si $Y_1\in\mathfrak{m}_{1,G_1-\mathrm{reg}}(F)$ et $Y_2\in\mathfrak{m}_{2,G_2-\mathrm{reg}}(F)$ se correspondent, alors

Let $(a,b)$ be a strongly regular semisimple collection of invariants with $a_i,b_i\in\mathcal{O}_E^{\sigma=(-1)^i}$, allowing arbitrary $b_0\in\mathcal{O}_F$. Let $\Delta_{a,b}$ b…