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Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local Gan-Gross-Prasad conjecture, we prove a local trace formula for the Ginzburg-Rallis model. By applying this trace formula, we prove the multiplicity one theorem for the Ginzburg-Rallis model over the tempered Vogan L-packets. In some cases, we also prove the epsilon dichotomy conjecture which gives a relation between the multiplicity and the exterior cube epsilon factor. This is a sequel work of [Wan15] in which we proved the geometric side of the trace formula.

Let $A$ be a finite dimensional k-algebra standardly stratified for a partial order $\leqslant$ and $\Delta$ be the direct sum of all standard modules. In this paper we study the extension algebra $E = Ext^{\ast}_A (Δ,Δ)$ of standard modules, characterize the stratification property of $E$ for $\geqslant$ and $\geqslant ^{op}$, and obtain a sufficient condition for $E$ to be a generalized Koszul algebra (in a sense which we define).

For a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$, Lusztig introduced the corresponding modified quantized enveloping algebra $\dot{\textbf{U}}$ and its canonical basis $\dot{\textbf{B}}$ in [13]. In this paper, in case $\mathfrak{g}$ is a symmetric Kac-Moody Lie algebra of finite or affine type, we define a set $\tilde{\mathcal{M}}$ which depends only on the root category $\mathcal{R}$ and prove that there is a bijection between $\tilde{\mathcal{M}}$ and $\dot{\textbf{B}}$, where $\mathcal{R}$ is the $T^2$-orbit category of the bounded derived category of corresponding Dynkin or tame quiver. Our method is based on a result of Lin, Xiao and Zhang in [10], which gives a PBW-type basis of $\textbf{U}^+$.

We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.