Let $F$ be a local field and $G= GL(n,F)$.

Assume that $\gamma$ is an element of $G$ and $G_\gamma$ is its centralizer. The orbital integral is defined as $$ O_\gamma^G( \phi) = \int\limits_{G_\gamma \backslash G} \phi( g^{-1} \gamma g) d g.$$

Assume that $\gamma$ is elliptic. Can we lift the elliptic orbital integral on $G$ to an elliptic orbital integral on $G' =GL(n,F')$, where $F' = F[X] / det(X-\gamma)$?

I think for explicit computations, it is much more convenient to work on $G'$, where $\gamma$ is diagonalizable.

What is the Reference? What is the Buzzword for this?

Let $F$ be a local field and $G= GL(n,F)$.

Assume that $\gamma$ is an element of $G$ and $G_\gamma$ is its centralizer. The orbital integral is defined as $$ O_\gamma^G( \phi) = \int\limits_{G_\gamma \backslash G} \phi( g^{-1} \gamma g) d g.$$

We can assume wlog that $\gamma$ is elliptic. Can we lift the elliptic orbital integral on $G$ to an elliptic orbital integral on $G' =GL(n,F')$, where $F' = F[X] / det(X-\gamma)$?

More precisely, can we find an $T:C_c^{\infty}(G) \rightarrow C_c^\infty(G')$ such that $$ O_\gamma^G(\phi) = O_\gamma^{G'} ( T\phi).$$

I think for explicit computations, it is much more convenient to work on $G'$, where $\gamma$ is diagonalizable nad $G'_\gamma$ can be chosen to be the diagonal matrices.

What is the Reference? What is the Buzzword for this?

Let $F$ be a local field and $G= GL(n,F)$.

Assume that $\gamma$ is an element of $G$ and $G_\gamma$ is its centralizer. The orbital integral is defined as $$ O_\gamma^G( \phi) = \int\limits_{G_\gamma \backslash G} \phi( g^{-1} \gamma g) d g.$$

Assume that $\gamma$ is elliptic. Can we lift the elliptic orbital integral on $G$ to an elliptic orbital integral on $G' =GL(n,F')$, where $F' = F[X] / det(X-\gamma)$? I think for explicit computations, it is much more convenient to work on $G'$. What is the Reference? What is the Buzzword for this?

Let $F$ be a local field and $G= GL(n,F)$.

Assume that $\gamma$ is an element of $G$ and $G_\gamma$ is its centralizer. The orbital integral is defined as $$ O_\gamma^G( \phi) = \int\limits_{G_\gamma \backslash G} \phi( g^{-1} \gamma g) d g.$$

Assume that $\gamma$ is elliptic. Can we lift the elliptic orbital integral on $G$ to an elliptic orbital integral on $G' =GL(n,F')$, where $F' = F[X] / det(X-\gamma)$?

I think for explicit computations, it is much more convenient to work on $G'$, where $\gamma$ is diagonalizable.

What is the Reference? What is the Buzzword for this?