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There was given the following definition of a Procesi bundle in this paper:

let $V$ denote a symplectic $\mathbb C$-vector space with a finite group $\Gamma$ of symplectic linear automorphisms and let $X \to V/\Gamma$ be a conical symplectic resolution.

Then a vector bundle $\mathcal F$ over $X$ is called a Procesi bundle if

(i) $\operatorname{End}_{\mathcal O_X} \mathcal F$ is $\mathbb C^*$-equivariantly isomorphic to $\mathbb C[V]\#\mathbb C[\Gamma]$;

(ii) $Ext^j(\mathcal F, \mathcal F) = 0$ for j >0.

Is it true that any fiber of Procesi bundle should be isomorphic as a $\Gamma$-module to the regular representation of $\Gamma$?

If so, how to prove this fact?

There was given the following definition of a Procesi bundle in this paper:

let $V$ denote a symplectic $\mathbb C$-vector space with a finite group $\Gamma$ of symplectic linear automorphisms and let $f: X \to V/\Gamma$ be a conical symplectic resolution.

Then a vector bundle $\mathcal F$ over $X$ is called a Procesi bundle if

(i) $\operatorname{End}_{\mathcal O_X} \mathcal F$ is $\mathbb C^*$-equivariantly isomorphic to $\mathbb C[V]\#\mathbb C[\Gamma]$;

(ii) $Ext^j(\mathcal F, \mathcal F) = 0$ for j >0.

Is it true that any fiber of Procesi bundle should be isomorphic as a $\Gamma$-module to the regular representation of $\Gamma$?

If so, how to prove this fact?

There was given the following definition of a Procesi bundle in this paper:

let $V$ denote a symplectic $\mathbb C$-vector space with a finite group $\Gamma$ of symplectic linear automorphisms and let $X \to V/\Gamma$ be a conical symplectic resolution.

Then a vector bundle $\mathcal F$ over $X$ is called a Procesi bundle if

(i) $\operatorname{End}_{\mathcal O_X} \mathcal F$ is $\mathbb C^*$-equivariantly isomorphic to $\mathbb C[V]\#\mathbb C[\Gamma]$;

(ii) $Ext^j(\mathcal F, \mathcal F) = 0$ for j >0.

Is it true that any fiber of Procesi bundle should be isomorphic as $\Gamma$-module to a regular representation of $\Gamma$?

If so, how to prove this fact?

There was given the following definition of a Procesi bundle in this paper:

let $V$ denote a symplectic $\mathbb C$-vector space with a finite group $\Gamma$ of symplectic linear automorphisms and let $X \to V/\Gamma$ be a conical symplectic resolution.

Then a vector bundle $\mathcal F$ over $X$ is called a Procesi bundle if

(i) $\operatorname{End}_{\mathcal O_X} \mathcal F$ is $\mathbb C^*$-equivariantly isomorphic to $\mathbb C[V]\#\mathbb C[\Gamma]$;

(ii) $Ext^j(\mathcal F, \mathcal F) = 0$ for j >0.

Is it true that any fiber of Procesi bundle should be isomorphic as a $\Gamma$-module to the regular representation of $\Gamma$?

If so, how to prove this fact?