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Let $\mathcal{O}_0$ be the principal block of the BGG category $\mathcal{O}$ for a finite dimensional simple Lie algebra over $\mathbb{C}$. For an element $w$ in the Weyl group $W$, let $\Delta_w$ denote the Verma module with highest weight $w_0w^{-1}\cdot 0$, where $w_0\in W$ is the longest element, and $\cdot$' denotes the dot-action.

It is well known that

$\mathrm{dim}(\mathrm{Hom}(\Delta_v, \Delta_w)) \leq 1$

This fact is pretty straightforward to prove algebraically. However, I do not know how to see this topologically. Namely, I do not know how to prove this via the interpretation of Verma modules as perverse sheaves on the flag variety.

I would be grateful if someone could explain how to see this fact topologically.

Added later: In response to Jim Humphreys comment let me add some motivation:

1. In this regard I think of category $\mathcal{O}$ as a "toy example". I would like to know what sort of generality this fact holds for. For instance, is the corresponding statement true for perverse sheaves smooth along a stratification given by affine spaces? The latter is certainly a highest weight category, computations in it can be undertaken topologically, etc. So as a starting point I would like to understand the topological reason for its truth for the "toy example".

2. In the same vein as 1) I would like to know whether this truly is a "geometric" fact, i.e., does it hold if I consider my sheaves with coefficients in a commutative ring say?

3. Computing extension groups of Verma modules is an old problem. If there is any hope for doing this topologically, I would think a reasonable place to start would be to compute $\mathrm{Ext}^0$ topologically!

4. In the same vein as 3). One can see that the extensions of Verma modules is given by (compactly supported) cohomology (appropriately shifted) of intersections of Schubert cells with opposite Schubert cells. This is related to my earlier questions:

Intersection of plus/minus cells in Bialynicki-Birula decomposition

A cohomology computation request.

• The above fact about homomorphisms between Vermas translates to the lowest non-vanishing cohomology (compactly supported) being one dimensional. These are smooth affine varieties, but (at least in low ranks) their Betti numbers satisfy a curious "Poincare duality"/palindromic type phenomenon. This phenomenon is even more starkly visible if one further looks at the Hodge numbers. Amusingly, since these varieties are smooth and irreducible, one immediately gets that the highest non-vanishing extension group (when it is possible to have morphisms between the Vermas) is one dimensional and concentrated in the "right" degree. This latter fact can also be shown algebraically, but requires a careful argument using translation functors (which can also be done geometrically without ever knowing anything about the intersections, but now I am digressing). Anyway, a topological reason as in my question may hopefully give some insight as to whether this palindromic phenomenon is a low rank coincidence or has any hope for holding in general.

Apologies if any of the reasons above are too vague/ranting, I didn't want to throw in all of that in my original question in case the answer was something blatantly obvious that I had been overlooking.

Let $\mathcal{O}_0$ be the principal block of the BGG category $\mathcal{O}$ for a finite dimensional simple Lie algebra over $\mathbb{C}$. For an element $w$ in the Weyl group $W$, let $\Delta_w$ denote the Verma module with highest weight $w_0w^{-1}\cdot 0$, where $w_0\in W$ is the longest element, and $\cdot$' denotes the dot-action.

It is well known that

$\mathrm{dim}(\mathrm{Hom}(\Delta_v, \Delta_w)) \leq 1$

This fact is pretty straightforward to prove algebraically. However, I do not know how to see this topologically. Namely, I do not know how to prove this via the interpretation of Verma modules as perverse sheaves on the flag variety.

I would be grateful if someone could explain how to see this fact topologically.

Added later: In response to Jim Humphreys comment let me add some motivation:

1. In this regard I think of category $\mathcal{O}$ as a "toy example". I would like to know what sort of generality this fact holds for. For instance, is the corresponding statement true for perverse sheaves smooth along a stratification given by affine spaces? The latter is certainly a highest weight category, computations in it can be undertaken topologically, etc. So as a starting point I would like to understand the topological reason for its truth for the "toy example".

2. In the same vein as 1) I would like to know whether this truly is a "geometric" fact, i.e., does it hold if I consider my sheaves with coefficients in a commutative ring say?

3. Computing extension groups of Verma modules is an old problem. If there is any hope for doing this topologically, I would think a reasonable place to start would be to compute $\mathrm{Ext}^0$ topologically!

4. In the same vein as 3). One can see that the extensions of Verma modules is given by (compactly supported) cohomology (appropriately shifted) of intersections of Schubert cells with opposite Schubert cells. This is related to my earlier questions:

Intersection of plus/minus cells in Bialynicki-Birula decomposition

A cohomology computation request.

• The above fact about homomorphisms between Vermas translates to the lowest non-vanishing cohomology (compactly supported) being one dimensional. These are smooth affine varieties, but (at least in low ranks) their Betti numbers satisfy a curious "Poincare duality"/palindromic type phenomenon. This phenomenon is even more starkly visible if one further looks at the Hodge numbers. Amusingly, since these varieties are smooth and irreducible, one immediately gets that the highest non-vanishing extension group (when it is possible to have morphisms between the Vermas) is one dimensional and concentrated in the "right" degree. This latter fact can also be shown algebraically, but requires a careful argument using translation functors (which can also be done geometrically without ever knowing anything about the intersections, but now I am digressing). Anyway, a topological reason as in my question may hopefully give some insight as to whether this palindromic phenomenon is a low rank coincidence or has any hope for holding in general.

Apologies if any of the reasons above are too vague/ranting, I didn't want to throw in all of that in my original question in case the answer was something blatantly obvious that I had been overlooking.

Let $\mathcal{O}_0$ be the principal block of the BGG category $\mathcal{O}$ for a finite dimensional simple Lie algebra over $\mathbb{C}$. For an element $w$ in the Weyl group $W$, let $\Delta_w$ denote the Verma module with highest weight $w_0w^{-1}\cdot 0$, where $w_0\in W$ is the longest element, and $\cdot$' denotes the dot-action.

It is well known that

$\mathrm{dim}(\mathrm{Hom}(\Delta_v, \Delta_w)) \leq 1$

This fact is pretty straightforward to prove algebraically. However, I do not know how to see this topologically. Namely, I do not know how to prove this via the interpretation of Verma modules as perverse sheaves on the flag variety.

I would be grateful if someone could explain how to see this fact topologically.

Added later: In response to Jim Humphreys comment let me add some motivation:

1. In this regard I think of category $\mathcal{O}$ as a "toy example". I would like to know what sort of generality this fact holds for. For instance, is the corresponding statement true for perverse sheaves smooth along a stratification given by affine spaces? The latter is certainly a highest weight category, computations in it can be undertaken topologically, etc. So as a starting point I would like to understand the topological reason for its truth for the "toy example".

2. In the same vein as 1) I would like to know whether this truly is a "geometric" fact, i.e., does it hold if I consider my sheaves with coefficients in a commutative ring say?

3. Computing extension groups of Verma modules is an old problem. If there is any hope for doing this topologically, I would think a reasonable place to start would be to compute $\mathrm{Ext}^0$ topologically!

4. In the same vein as 4). One can see that the extensions of Verma modules is given by (compactly supported) cohomology (appropriately shifted) of intersections of Schubert cells with opposite Schubert cells. This is related to my earlier questions:

Intersection of plus/minus cells in Bialynicki-Birula decomposition

A cohomology computation request.

• The above fact about homomorphisms between Vermas translates to the lowest non-vanishing cohomology (compactly supported) being one dimensional. These are smooth affine varieties, but (at least in low ranks) their Betti numbers satisfy a curious "Poincare duality"/palindromic type phenomenon. This phenomenon is even more starkly visible if one further looks at the Hodge numbers. Amusingly, since these varieties are smooth and irreducible, one immediately gets that the highest non-vanishing extension group (when it is possible to have morphisms between the Vermas) is one dimensional and concentrated in the "right" degree. This latter fact can also be shown algebraically, but requires a careful argument using translation functors (which can also be done geometrically without ever knowing anything about the intersections, but now I am digressing). Anyway, a topological reason as in my question may hopefully give some insight as to whether this palindromic phenomenon is a low rank coincidence or has any hope for holding in general.

Apologies if any of the reasons above are too vague/ranting, I didn't want to throw in all of that in my original question in case the answer was something blatantly obvious that I had been overlooking.

Let $\mathcal{O}_0$ be the principal block of the BGG category $\mathcal{O}$ for a finite dimensional simple Lie algebra over $\mathbb{C}$. For an element $w$ in the Weyl group $W$, let $\Delta_w$ denote the Verma module with highest weight $w_0w^{-1}\cdot 0$, where $w_0\in W$ is the longest element, and $\cdot$' denotes the dot-action.

It is well known that

$\mathrm{dim}(\mathrm{Hom}(\Delta_v, \Delta_w)) \leq 1$

This fact is pretty straightforward to prove algebraically. However, I do not know how to see this topologically. Namely, I do not know how to prove this via the interpretation of Verma modules as perverse sheaves on the flag variety.

I would be grateful if someone could explain how to see this fact topologically.

Added later: In response to Jim Humphreys comment let me add some motivation:

1. In this regard I think of category $\mathcal{O}$ as a "toy example". I would like to know what sort of generality this fact holds for. For instance, is the corresponding statement true for perverse sheaves smooth along a stratification given by affine spaces? The latter is certainly a highest weight category, computations in it can be undertaken topologically, etc. So as a starting point I would like to understand the topological reason for its truth for the "toy example".

2. In the same vein as 1) I would like to know whether this truly is a "geometric" fact, i.e., does it hold if I consider my sheaves with coefficients in a commutative ring say?

3. Computing extension groups of Verma modules is an old problem. If there is any hope for doing this topologically, I would think a reasonable place to start would be to compute $\mathrm{Ext}^0$ topologically!

4. In the same vein as 3). One can see that the extensions of Verma modules is given by (compactly supported) cohomology (appropriately shifted) of intersections of Schubert cells with opposite Schubert cells. This is related to my earlier questions:

Intersection of plus/minus cells in Bialynicki-Birula decomposition

A cohomology computation request.

• The above fact about homomorphisms between Vermas translates to the lowest non-vanishing cohomology (compactly supported) being one dimensional. These are smooth affine varieties, but (at least in low ranks) their Betti numbers satisfy a curious "Poincare duality"/palindromic type phenomenon. This phenomenon is even more starkly visible if one further looks at the Hodge numbers. Amusingly, since these varieties are smooth and irreducible, one immediately gets that the highest non-vanishing extension group (when it is possible to have morphisms between the Vermas) is one dimensional and concentrated in the "right" degree. This latter fact can also be shown algebraically, but requires a careful argument using translation functors (which can also be done geometrically without ever knowing anything about the intersections, but now I am digressing). Anyway, a topological reason as in my question may hopefully give some insight as to whether this palindromic phenomenon is a low rank coincidence or has any hope for holding in general.

Apologies if any of the reasons above are too vague/ranting, I didn't want to throw in all of that in my original question in case the answer was something blatantly obvious that I had been overlooking.

Let $\mathcal{O}_0$ be the principal block of the BGG category $\mathcal{O}$ for a finite dimensional simple Lie algebra over $\mathbb{C}$. For an element $w$ in the Weyl group $W$, let $\Delta_w$ denote the Verma module with highest weight $w_0w^{-1}\cdot 0$, where $w_0\in W$ is the longest element, and $\cdot$' denotes the dot-action.

It is well known that

$\mathrm{dim}(\mathrm{Hom}(\Delta_v, \Delta_w)) \leq 1$

This fact is pretty straightforward to prove algebraically. However, I do not know how to see this topologically. Namely, I do not know how to prove this via the interpretation of Verma modules as perverse sheaves on the flag variety.

I would be grateful if someone could explain how to see this fact topologically.

Let $\mathcal{O}_0$ be the principal block of the BGG category $\mathcal{O}$ for a finite dimensional simple Lie algebra over $\mathbb{C}$. For an element $w$ in the Weyl group $W$, let $\Delta_w$ denote the Verma module with highest weight $w_0w^{-1}\cdot 0$, where $w_0\in W$ is the longest element, and $\cdot$' denotes the dot-action.

It is well known that

$\mathrm{dim}(\mathrm{Hom}(\Delta_v, \Delta_w)) \leq 1$

This fact is pretty straightforward to prove algebraically. However, I do not know how to see this topologically. Namely, I do not know how to prove this via the interpretation of Verma modules as perverse sheaves on the flag variety.

I would be grateful if someone could explain how to see this fact topologically.

Added later: In response to Jim Humphreys comment let me add some motivation:

1. In this regard I think of category $\mathcal{O}$ as a "toy example". I would like to know what sort of generality this fact holds for. For instance, is the corresponding statement true for perverse sheaves smooth along a stratification given by affine spaces? The latter is certainly a highest weight category, computations in it can be undertaken topologically, etc. So as a starting point I would like to understand the topological reason for its truth for the "toy example".

2. In the same vein as 1) I would like to know whether this truly is a "geometric" fact, i.e., does it hold if I consider my sheaves with coefficients in a commutative ring say?

3. Computing extension groups of Verma modules is an old problem. If there is any hope for doing this topologically, I would think a reasonable place to start would be to compute $\mathrm{Ext}^0$ topologically!

4. In the same vein as 4). One can see that the extensions of Verma modules is given by (compactly supported) cohomology (appropriately shifted) of intersections of Schubert cells with opposite Schubert cells. This is related to my earlier questions:

Intersection of plus/minus cells in Bialynicki-Birula decomposition

A cohomology computation request.

• The above fact about homomorphisms between Vermas translates to the lowest non-vanishing cohomology (compactly supported) being one dimensional. These are smooth affine varieties, but (at least in low ranks) their Betti numbers satisfy a curious "Poincare duality"/palindromic type phenomenon. This phenomenon is even more starkly visible if one further looks at the Hodge numbers. Amusingly, since these varieties are smooth and irreducible, one immediately gets that the highest non-vanishing extension group (when it is possible to have morphisms between the Vermas) is one dimensional and concentrated in the "right" degree. This latter fact can also be shown algebraically, but requires a careful argument using translation functors (which can also be done geometrically without ever knowing anything about the intersections, but now I am digressing). Anyway, a topological reason as in my question may hopefully give some insight as to whether this palindromic phenomenon is a low rank coincidence or has any hope for holding in general.

Apologies if any of the reasons above are too vague/ranting, I didn't want to throw in all of that in my original question in case the answer was something blatantly obvious that I had been overlooking.