35 problems

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 $\mathbf{G}$ and $\mathbf{H}$ be reductive groups, set $G=\mathbf{G}(F)$ and $H=\mathbf{H}(F)$, and let $H'$ be the subset of $\mathbf{G}$-regular elements of $H$. For a charac…

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 ${\mathfrak g}=\operatorname{so}(2n+1)$, and let $\operatorname{strip}(r)$ be the set of all $X\in{\mathfrak g}$ such that $|\alpha(X)|=q^{-r}$ for every root $\alpha$. For…

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 $K=\mathbb{Q}$, 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…

Arithmetic Fundamental Lemma. For every regular semi-simple element $\gamma\in G'_{\mathrm{rs}}$,

Let $f'_D$ be the normalized test function associated with the standard lattice chain, and let $f'_{\mathrm{corr}}\in C_c^\infty(G')$ be a correction function. For…

Let $f_D=1_{O_D^\times}\in C_c^\infty(G)$ be the standard test function. A transfer $f”_D\in C_c^\infty(G')$ is a function whose orbital integrals match those of $f_D$ on matching…

Let $f_D=1_{O_D^\times}\in C_c^\infty(G)$ be the standard test function, and let $f”_D\in C_c^\infty(G')$ be a transfer of $f_D$. For $[b]\in B(H,\mu_H)$ and regular semi-simple…

Let $G'_{\mathrm{rs},W_1}$ be the open subset of $\operatorname{GL}_n(F)\times\operatorname{GL}_{n+1}(F)$ consisting of regular semisimple elements matching with elements in the no…

Let $G_n={\mathrm{GL}}_n(F)$, where $F$ is a $p$-adic local field, let $C_c^{\infty}(G_n)$ denote the compactly supported locally constant functions on $G_n$, and let…

Let $F$ be the base local field, let $K_0,K_1,K_2,K_3$ be the quadratic $F$-algebras arising from a biquadratic extension, and let … be $F$-algebra embeddings. Assume that…

Assume the signatures of $V$ at the two archimedean places of $E_0$ are $((n-1,1),(n,0))$, so that the associated CM cycle has been defined in the relevant Lubin--Tate space. Let t…

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…

Assume that $F/F_0$ is unramified, the special vectors $u_i$ have norm one, and use the same Haar measure and transfer factor as in the unramified fundamental lemma. Let…

Let $F/F_0$ be a quadratic extension, let $S_n(F_0)$ be the symmetric space, and let $W_0,W_1$ be the relevant hermitian spaces. For $f'\in C_c^\infty(S_n(F_0))$, a transfer is a p…