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Since you are posing the question for reductive $G$, you should work with the Iwahori Weyl group $W_{I}$ as this is what parametrizes $I\backslash G/ I$. However since the question is only about volumes, the answer is independent if $ w $ in the affine Weyl group is perturbed by a length zero element.

To write the volume, you have to replace $w$ by the unique element in $W_{P}wW_{P}$ that has minimal length, where $W_{P}$ is the Weyl group of $P$. I assume that $w$ is so chosen. Then $ W_{P} \cap w W_{P} w^{-1} $ is Coxeter subgroup of $W_{P}$ and each coset in $ W_{w,P} := W_{P} / (W \cap w W_{P} w^{-1})$ also has a unique element of minimal possible length. Let $S_{w,P} \subset W_{P}$ denote the representative set of these minimal length elements. Then $$ \mu ( P w P ) = \sum_{\sigma \in S_{w,P}} q^{l(\sigma w)} \mu(P) $$

Edit: for such $\sigma, w$, $l(\sigma w) = l(\sigma) + l(w) $.

Since you are posing the question for reductive $G$, you should work with the Iwahori Weyl group $W_{I}$ as this is what parametrizes $I\backslash G/ I$. However since the question is only about volumes, the answer is independent if $ w $ in the affine Weyl group is perturbed by a length zero element.

To write the volume, you have to replace $w$ by the unique element in $W_{P}wW_{P}$ that has minimal length, where $W_{P}$ is the Weyl group of $P$. I assume that $w$ is so chosen. Then $ W_{P} \cap w W_{P} w^{-1} $ is Coxeter subgroup of $W_{P}$ and each coset in $ W_{w,P} := W_{P} / (W \cap w W_{P} w^{-1})$ also has a unique element of minimal possible length. Let $S_{w,P} \subset W_{P}$ denote the representative set of these minimal length elements. Then $$ \mu ( P w P ) = \sum_{\sigma \in S_{w,P}} q^{l(\sigma w)} \mu(P) $$

Since you are posing the question for reductive $G$, you should work with the Iwahori Weyl group $W_{I}$ as this is what parametrizes $I\backslash G/ I$. However since the question is only about volumes, the answer is independent if $ w $ in the affine Weyl group is perturbed by a length zero element.

To write the volume, you have to replace $w$ by the unique element in $W_{P}wW_{P}$ that has minimal length, where $W_{P}$ is the Weyl group of $P$. I assume that $w$ is so chosen. Then $ W_{P} \cap w W_{P} w^{-1} $ is Coxeter subgroup of $W_{P}$ and each coset in $ W_{w,P} := W_{P} / (W \cap w W_{P} w^{-1})$ also has a unique element of minimal possible length. Let $S_{w,P} \subset W_{P}$ denote the representative set of these minimal length elements. Then $$ \mu ( P w P ) = \sum_{\sigma \in S_{w,P}} q^{l(\sigma w)} \mu(P) $$

Edit: for such $\sigma, w$, $l(\sigma w) = l(\sigma) + l(w) $.

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Hodge-Tate
Hodge-Tate
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blah blah uselessSince you are posing the question for reductive $G$, you should work with the Iwahori Weyl group $W_{I}$ as this is what parametrizes $I\backslash G/ I$. However since the question is only about volumes, the answer doesn't contain any useful information blah blahis independent if $ w $ in the affine Weyl group is perturbed by a length zero element.

To write the volume, you have to replace $w$ by the unique element in $W_{P}wW_{P}$ that has minimal length, where $W_{P}$ is the Weyl group of $P$. I assume that $w$ is so chosen. Then $ W_{P} \cap w W_{P} w^{-1} $ is Coxeter subgroup of $W_{P}$ and each coset in $ W_{w,P} := W_{P} / (W \cap w W_{P} w^{-1})$ also has a unique element of minimal possible length. Let $S_{w,P} \subset W_{P}$ denote the representative set of these minimal length elements. Then $$ \mu ( P w P ) = \sum_{\sigma \in S_{w,P}} q^{l(\sigma w)} \mu(P) $$

blah blah useless answer doesn't contain any useful information blah blah

Since you are posing the question for reductive $G$, you should work with the Iwahori Weyl group $W_{I}$ as this is what parametrizes $I\backslash G/ I$. However since the question is only about volumes, the answer is independent if $ w $ in the affine Weyl group is perturbed by a length zero element.

To write the volume, you have to replace $w$ by the unique element in $W_{P}wW_{P}$ that has minimal length, where $W_{P}$ is the Weyl group of $P$. I assume that $w$ is so chosen. Then $ W_{P} \cap w W_{P} w^{-1} $ is Coxeter subgroup of $W_{P}$ and each coset in $ W_{w,P} := W_{P} / (W \cap w W_{P} w^{-1})$ also has a unique element of minimal possible length. Let $S_{w,P} \subset W_{P}$ denote the representative set of these minimal length elements. Then $$ \mu ( P w P ) = \sum_{\sigma \in S_{w,P}} q^{l(\sigma w)} \mu(P) $$

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Hodge-Tate
Hodge-Tate
  • 171
  • 5

Since you are posing the question for reductive $G$, you should work with the Iwahori Weyl group $W_{I}$ as this is what parametrizes $I\backslash G/ I$. However since the question is only about volumes, theblah blah useless answer is independent if $ w $ in the affine Weyl group is perturbed by a length zero element.

To write the volume, you have to replace $w$ by the unique element in $W_{P}wW_{P}$ that has minimal length, where $W_{P}$ is the Weyl group of $P$. I assume that $w$ is so chosen. Then $ W_{P} \cap w W_{P} w^{-1} $ is Coxeter subgroup of $W_{P}$ and each coset in $ W_{w,P} := W_{P} / (W \cap w W_{P} w^{-1})$ also has a unique element of minimal possible length. Let $S_{w,P} \subset W_{P}$ denote the representative set of these minimal length elements. Then $$ \mu ( P w P ) = \sum_{\sigma \in S_{w,P}} q^{l(\sigma w)} \mu(P) $$doesn't contain any useful information blah blah

Since you are posing the question for reductive $G$, you should work with the Iwahori Weyl group $W_{I}$ as this is what parametrizes $I\backslash G/ I$. However since the question is only about volumes, the answer is independent if $ w $ in the affine Weyl group is perturbed by a length zero element.

To write the volume, you have to replace $w$ by the unique element in $W_{P}wW_{P}$ that has minimal length, where $W_{P}$ is the Weyl group of $P$. I assume that $w$ is so chosen. Then $ W_{P} \cap w W_{P} w^{-1} $ is Coxeter subgroup of $W_{P}$ and each coset in $ W_{w,P} := W_{P} / (W \cap w W_{P} w^{-1})$ also has a unique element of minimal possible length. Let $S_{w,P} \subset W_{P}$ denote the representative set of these minimal length elements. Then $$ \mu ( P w P ) = \sum_{\sigma \in S_{w,P}} q^{l(\sigma w)} \mu(P) $$

blah blah useless answer doesn't contain any useful information blah blah

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Hodge-Tate
Hodge-Tate
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Hodge-Tate
Hodge-Tate
  • 171
  • 5