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Let $\mathfrak{g}$ be a simple complex Lie algebra, and $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra with Weyl group $W$. Consider the fibre product $\mathfrak{h}\times_{\mathfrak{g}} N$, where $N\subset \mathfrak{g}$ is the nilpotent cone. It is a zero-dimensional affine $\mathbb{C}$-scheme with coordinate ring $S/(S_{+}^{W})$, where $S=\mathbb{C}[\mathfrak{h}^{\ast}]$ is the coordinate ring of $\mathfrak{h}^{\ast}$ and $(S_{+}^{W})$ is the ideal generated by the $W$-invariants in $\mathbb{C}[\mathfrak{h}^{\ast}]$ with no constant term: the ring $S/(S_{+}^{W})$ is known as the algebra of coinvariants; it is a finite dimensional $W$-module. In fact, it is isomorphic (as a graded $W$-module) to the cohomology of the flag variety.

Now, let $\lambda$ be a dominant weight lying in the root lattice. Then, the corresponding finite dimensional $\mathfrak{g}$-module $L(\lambda)$ admits a zero weight space $L(\lambda)[0]$; moreover, since $W$ permutes weight spaces we have that this zero weight space is $W$-invariant, hence a $W$-module. Now set $S=\text{Sym}(L(\lambda)[0])$, the symmetric algebra of $L(\lambda)[0]$ and define $S_{+}^{W}$ to be the set of $W$-invariants without constant term. Then, $R_{\lambda} \stackrel{def}{=}S/(S_{+}^{W})$ is a finite dimensional $W$-module (since this is the coordinate ring of the scheme theoretic fibre of the image of $0$ in the quotient $L(\lambda)[0]^{\ast}//W$). Hence, to each dominant root lattice element $\lambda$ of $\mathfrak{g}$ we have associated a finite dimensional $W$-module $R_{\lambda}$.

Question(s): What is known about these $W$-modules $R_{\lambda}$? Do they arise 'naturally' anywhere? (eg, if $\lambda$ is the longest root of $\mathfrak{g}$ then $R_{\lambda}$ is the cohomology of the flag variety).

Any references or pointers to what's 'really' going on would be much appreciated.

I should also point out that in type $A_{n}$ it is known that when

$$\lambda=(\mu_{1}-\mu_{2})\omega_{1}+ (\mu_{2}-\mu_{3})\omega_{2}+\ldots + \mu_{n}\omega_{n}$$

and $\mu_{1}+\ldots + \mu_{n}=n+1$ then $L(\lambda)[0]$ is the simple $S_{n+1}$-module $V^{\mu'}$, where $\mu'$ is the dual partition of $\mu: \mu_{1}\geq \mu_{2} \geq \ldots \mu_{n}\geq 0$.

Let $\mathfrak{g}$ be a simple complex Lie algebra, and $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra with Weyl group $W$. Consider the fibre product $\mathfrak{h}\times_{\mathfrak{g}} N$, where $N\subset \mathfrak{g}$ is the nilpotent cone. It is a zero-dimensional affine $\mathbb{C}$-scheme with coordinate ring $S/(S_{+}^{W})$, where $S=\mathbb{C}[\mathfrak{h}^{\ast}]$ is the coordinate ring of $\mathfrak{h}^{\ast}$ and $(S_{+}^{W})$ is the ideal generated by the $W$-invariants in $\mathbb{C}[\mathfrak{h}^{\ast}]$ with no constant term: the ring $S/(S_{+}^{W})$ is known as the algebra of coinvariants; it is a finite dimensional $W$-module. In fact, it is isomorphic (as a graded $W$-module) to the cohomology of the flag variety.

Now, let $\lambda$ be a dominant weight lying in the root lattice. Then, the corresponding finite dimensional $\mathfrak{g}$-module $L(\lambda)$ admits a zero weight space $L(\lambda)[0]$; moreover, since $W$ permutes weight spaces we have that this zero weight space is $W$-invariant, hence a $W$-module. Now set $S=\text{Sym}(L(\lambda)[0])$, the symmetric algebra of $L(\lambda)[0]$ and define $S_{+}^{W}$ to be the set of $W$-invariants without constant term. Then, $R_{\lambda} \stackrel{def}{=}S/(S_{+}^{W})$ is a finite dimensional $W$-module (since this is the coordinate ring of the scheme theoretic fibre of the image of $0$ in the quotient $L(\lambda)[0]^{\ast}//W$). Hence, to each dominant root lattice element $\lambda$ of $\mathfrak{g}$ we have associated a finite dimensional $W$-module $R_{\lambda}$.

Question(s): What is known about these $W$-modules $R_{\lambda}$? Do they arise 'naturally' anywhere? (eg, if $\lambda$ is the highest root of $\mathfrak{g}$ then $R_{\lambda}$ is the cohomology of the flag variety).

Any references or pointers to what's 'really' going on would be much appreciated.

I should also point out that in type $A_{n}$ it is known that when

$$\lambda=(\mu_{1}-\mu_{2})\omega_{1}+ (\mu_{2}-\mu_{3})\omega_{2}+\ldots + \mu_{n}\omega_{n}$$

and $\mu_{1}+\ldots + \mu_{n}=n+1$ then $L(\lambda)[0]$ is the simple $S_{n+1}$-module $V^{\mu'}$, where $\mu'$ is the dual partition of $\mu: \mu_{1}\geq \mu_{2} \geq \ldots \mu_{n}\geq 0$.

Let $\mathfrak{g}$ be a simple complex Lie algebra, and $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra with Weyl group $W$. Consider the fibre product $\mathfrak{h}\times_{\mathfrak{g}} N$, where $N\subset \mathfrak{g}$ is the nilpotent cone. It is a zero-dimensional affine $\mathbb{C}$-scheme with coordinate ring $S/(S_{+}^{W})$, where $S=\mathbb{C}[\mathfrak{h}^{\ast}]$ is the coordinate ring of $\mathfrak{h}^{\ast}$ and $(S_{+}^{W})$ is the ideal generated by the $W$-invariants in $\mathbb{C}[\mathfrak{h}^{\ast}]$ with no constant term: the ring $S/(S_{+}^{W})$ is known as the algebra of coinvariants; it is a finite dimensional $W$-module. In fact, it is isomorphic (as a graded $W$-module) to the cohomology of the flag variety.

Now, let $\lambda$ be a dominant weight lying in the root lattice. Then, the corresponding finite dimensional $\mathfrak{g}$-module $L(\lambda)$ admits a zero weight space $L(\lambda)[0]$; moreover, since $W$ permutes weight spaces we have that this zero weight space is $W$-invariant, hence a $W$-module. Now set $S=\text{Sym}(L(\lambda)[0])$, the symmetric algebra of $L(\lambda)[0]$ and define $S_{+}^{W}$ to be the set of $W$-invariants without constant term. Then, $R_{\lambda} \stackrel{def}{=}S/(S_{+}^{W})$ is a finite dimensional $W$-module (since this is the coordinate ring of the scheme theoretic fibre of the image of $0$ in the quotient $L(\lambda)[0]^{\ast}//W$). Hence, to each dominant root $\lambda$ of $\mathfrak{g}$ we have associated a finite dimensional $W$-module $R_{\lambda}$.

Question(s): What is known about these $W$-modules $R_{\lambda}$? Do they arise 'naturally' anywhere? (eg, if $\lambda$ is the longest root of $\mathfrak{g}$ then $R_{\lambda}$ is the cohomology of the flag variety).

Any references or pointers to what's 'really' going on would be much appreciated.

I should also point out that in type $A_{n}$ it is known that when

$$\lambda=(\mu_{1}-\mu_{2})\omega_{1}+ (\mu_{2}-\mu_{3})\omega_{2}+\ldots + \mu_{n}\omega_{n}$$

and $\mu_{1}+\ldots + \mu_{n}=n+1$ then $L(\lambda)[0]$ is the simple $S_{n+1}$-module $V^{\mu'}$, where $\mu'$ is the dual partition of $\mu: \mu_{1}\geq \mu_{2} \geq \ldots \mu_{n}\geq 0$.

Let $\mathfrak{g}$ be a simple complex Lie algebra, and $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra with Weyl group $W$. Consider the fibre product $\mathfrak{h}\times_{\mathfrak{g}} N$, where $N\subset \mathfrak{g}$ is the nilpotent cone. It is a zero-dimensional affine $\mathbb{C}$-scheme with coordinate ring $S/(S_{+}^{W})$, where $S=\mathbb{C}[\mathfrak{h}^{\ast}]$ is the coordinate ring of $\mathfrak{h}^{\ast}$ and $(S_{+}^{W})$ is the ideal generated by the $W$-invariants in $\mathbb{C}[\mathfrak{h}^{\ast}]$ with no constant term: the ring $S/(S_{+}^{W})$ is known as the algebra of coinvariants; it is a finite dimensional $W$-module. In fact, it is isomorphic (as a graded $W$-module) to the cohomology of the flag variety.

Now, let $\lambda$ be a dominant weight lying in the root lattice. Then, the corresponding finite dimensional $\mathfrak{g}$-module $L(\lambda)$ admits a zero weight space $L(\lambda)[0]$; moreover, since $W$ permutes weight spaces we have that this zero weight space is $W$-invariant, hence a $W$-module. Now set $S=\text{Sym}(L(\lambda)[0])$, the symmetric algebra of $L(\lambda)[0]$ and define $S_{+}^{W}$ to be the set of $W$-invariants without constant term. Then, $R_{\lambda} \stackrel{def}{=}S/(S_{+}^{W})$ is a finite dimensional $W$-module (since this is the coordinate ring of the scheme theoretic fibre of the image of $0$ in the quotient $L(\lambda)[0]^{\ast}//W$). Hence, to each dominant root lattice element $\lambda$ of $\mathfrak{g}$ we have associated a finite dimensional $W$-module $R_{\lambda}$.

Question(s): What is known about these $W$-modules $R_{\lambda}$? Do they arise 'naturally' anywhere? (eg, if $\lambda$ is the longest root of $\mathfrak{g}$ then $R_{\lambda}$ is the cohomology of the flag variety).

Any references or pointers to what's 'really' going on would be much appreciated.

I should also point out that in type $A_{n}$ it is known that when

$$\lambda=(\mu_{1}-\mu_{2})\omega_{1}+ (\mu_{2}-\mu_{3})\omega_{2}+\ldots + \mu_{n}\omega_{n}$$

and $\mu_{1}+\ldots + \mu_{n}=n+1$ then $L(\lambda)[0]$ is the simple $S_{n+1}$-module $V^{\mu'}$, where $\mu'$ is the dual partition of $\mu: \mu_{1}\geq \mu_{2} \geq \ldots \mu_{n}\geq 0$.

Let $\mathfrak{g}$ be a simple complex Lie algebra, and $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra with Weyl group $W$. Consider the fibre product $\mathfrak{h}\times_{\mathfrak{g}} N$, where $N\subset \mathfrak{g}$ is the nilpotent cone. It is a zero-dimensional affine $\mathbb{C}$-scheme with coordinate ring $S/(S_{+}^{W})$, where $S=\mathbb{C}[\mathfrak{h}^{\ast}]$ is the coordinate ring of $\mathfrak{h}^{\ast}$ and $(S_{+}^{W})$ is the ideal generated by the $W$-invariants in $\mathbb{C}[\mathfrak{h}^{\ast}]$ with no constant term: the ring $S/(S_{+}^{W})$ is known as the algebra of coinvariants; it is a finite dimensional $W$-module. In fact, it is isomorphic (as a graded $W$-module) to the cohomology of the flag variety.

Now, let $\lambda$ be a dominant weight lying in the root lattice. Then, the corresponding finite dimensional $\mathfrak{g}$-module $L(\lambda)$ admits a zero weight space $L(\lambda)[0]$; moreover, since $W$ permutes weight spaces we have that this zero weight space is $W$-invariant, hence a $W$-module. Now set $S=\text{Sym}(L(\lambda)[0])$, the symmetric algebra of $L(\lambda)[0]$ and define $S_{+}^{W}$ to be the set of $W$-invariants without constant term. Then, $R_{\lambda} \stackrel{def}{=}S/(S_{+}^{W})$ is a finite dimensional $W$-module (since this is the coordinate ring of the scheme theoretic fibre of the image of $0$ in the quotient $L(\lambda)[0]//W$). Hence, to each dominant root $\lambda$ of $\mathfrak{g}$ we have associated a finite dimensional $W$-module $R_{\lambda}$.

Question(s): What is known about these $W$-modules $R_{\lambda}$? Do they arise 'naturally' anywhere? (eg, if $\lambda$ is the longest root of $\mathfrak{g}$ then $R_{\lambda}$ is the cohomology of the flag variety).

Any references or pointers to what's 'really' going on would be much appreciated.

I should also point out that in type $A_{n}$ it is known that when

$$\lambda=(\mu_{1}-\mu_{2})\omega_{1}+ (\mu_{2}-\mu_{3})\omega_{2}+\ldots + \mu_{n}\omega_{n}$$

and $\mu_{1}+\ldots + \mu_{n}=n+1$ then $L(\lambda)[0]$ is the simple $S_{n+1}$-module $V^{\mu'}$, where $\mu'$ is the dual partition of $\mu: \mu_{1}\geq \mu_{2} \geq \ldots \mu_{n}\geq 0$.

Let $\mathfrak{g}$ be a simple complex Lie algebra, and $\mathfrak{h}\subset\mathfrak{g}$ a Cartan subalgebra with Weyl group $W$. Consider the fibre product $\mathfrak{h}\times_{\mathfrak{g}} N$, where $N\subset \mathfrak{g}$ is the nilpotent cone. It is a zero-dimensional affine $\mathbb{C}$-scheme with coordinate ring $S/(S_{+}^{W})$, where $S=\mathbb{C}[\mathfrak{h}^{\ast}]$ is the coordinate ring of $\mathfrak{h}^{\ast}$ and $(S_{+}^{W})$ is the ideal generated by the $W$-invariants in $\mathbb{C}[\mathfrak{h}^{\ast}]$ with no constant term: the ring $S/(S_{+}^{W})$ is known as the algebra of coinvariants; it is a finite dimensional $W$-module. In fact, it is isomorphic (as a graded $W$-module) to the cohomology of the flag variety.

Now, let $\lambda$ be a dominant weight lying in the root lattice. Then, the corresponding finite dimensional $\mathfrak{g}$-module $L(\lambda)$ admits a zero weight space $L(\lambda)[0]$; moreover, since $W$ permutes weight spaces we have that this zero weight space is $W$-invariant, hence a $W$-module. Now set $S=\text{Sym}(L(\lambda)[0])$, the symmetric algebra of $L(\lambda)[0]$ and define $S_{+}^{W}$ to be the set of $W$-invariants without constant term. Then, $R_{\lambda} \stackrel{def}{=}S/(S_{+}^{W})$ is a finite dimensional $W$-module (since this is the coordinate ring of the scheme theoretic fibre of the image of $0$ in the quotient $L(\lambda)[0]^{\ast}//W$). Hence, to each dominant root $\lambda$ of $\mathfrak{g}$ we have associated a finite dimensional $W$-module $R_{\lambda}$.

Question(s): What is known about these $W$-modules $R_{\lambda}$? Do they arise 'naturally' anywhere? (eg, if $\lambda$ is the longest root of $\mathfrak{g}$ then $R_{\lambda}$ is the cohomology of the flag variety).

Any references or pointers to what's 'really' going on would be much appreciated.

I should also point out that in type $A_{n}$ it is known that when

$$\lambda=(\mu_{1}-\mu_{2})\omega_{1}+ (\mu_{2}-\mu_{3})\omega_{2}+\ldots + \mu_{n}\omega_{n}$$

and $\mu_{1}+\ldots + \mu_{n}=n+1$ then $L(\lambda)[0]$ is the simple $S_{n+1}$-module $V^{\mu'}$, where $\mu'$ is the dual partition of $\mu: \mu_{1}\geq \mu_{2} \geq \ldots \mu_{n}\geq 0$.