Projective Functors and the Soergel Functor in Category O — Examples

Assume the Axiom of Choice, inherited from the published items used. We work over $\mathbb{k}=\mathbb{C}$ and compute the coinvariant algebra of The coinvariant algebra of a finite Weyl group in the two smallest cases and read off its graded decomposition, checking the general theorem The coinvariant algebra is the graded regular representation of W; the example is a verification in small rank and is not used as evidence for the general statement.

The case $W=S_2$. Here $r=1$, the auxiliary polynomial ring is $\mathbb{k}[x]$ with the transposition acting by $x\mapsto -x$, and the basic invariant is $x^2$, so $C=\mathbb{k}[x]\mathrm{/}(x^2)$ with $\mathrm{deg}x=2$ in the page normalization and $\mathrm{deg}x=1$ for the classical Hilbert series. The Hilbert series is therefore $1+t^2$ in page degrees (equivalently $1+t$ in classical degrees), agreeing with ${\prod }_i(1+t^2+\cdots +t^{2(d_i-1)})$ for the single basic degree $d_1=2$ of Weyl coinvariant hilbert series has order w dimension. The degree-zero piece $C_0=\mathbb{k}$ is the trivial representation, and the one-dimensional top piece $C_2=\mathbb{k}x$ is the sign representation, so $C$ is the regular representation of $S_2$ with graded multiplicities one for the trivial and one for the sign module.

The case $W=S_3$. Use the auxiliary permutation model of The coinvariant algebra of a finite Weyl group: $C=\mathbb{k}[x_1,x_2,x_3]\mathrm{/}(e_1,e_2,e_3),e_1=x_1+x_2+x_3,e_2=x_1{x}_2+x_1{x}_3+x_2{x}_3,e_3=x_1{x}_2{x}_3,$ with $S_3$ permuting the variables. In classical degrees the basic degrees are $1,2,3$, so the Hilbert series is $\frac{(1-t)(1-t^2)(1-t^3)}{(1-t{)}^3}=(1+t)(1+t+t^2)=1+2t+2t^2+t^3,$ and in the page normalization, where each variable has degree $2$, it becomes $(1+t^2)(1+t^2+t^4)=1+2t^2+2t^4+t^6$. The graded pieces have dimensions $1,2,2,1$ in degrees $0,2,4,6$.

Graded pieces as $S_3$-modules. Identify $C$ with the coinvariant algebra of the two-dimensional reflection representation $\mathfrak{h}\subset {\mathbb{k}}^3$. The degree-zero piece $C_0$ is the trivial module. The degree-two piece $C_2$ is spanned by the images of $x_1,x_2,x_3$ modulo $e_1$; these span a two-dimensional space on which $S_3$ acts as the standard reflection representation. Put $a=x_1$, $b=x_2$, $x_3=-a-b$. The relations are $a^2+ab+b^2=0$ and $a^{2}b+a{b}^2=0$, so $b^3=0$ and $C_6$ is spanned by the nonzero class $a{b}^2$ (its nonvanishing also follows from the Hilbert series). The Vandermonde reduces to $-6a{b}^2$, so it spans $C_6$ and carries the sign character. The perfect multiplication pairing $C_2\times C_4\to C_6$ from The coinvariant algebra is a complete intersection with one-dimensional socle identifies $C_4$ with $C_2^{\ast }\otimes \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}$. The standard reflection representation is self-dual: the invariant coordinate dot product restricts nondegenerately to the sum-zero plane. Hence $C_4$ is standard tensor sign. Thus $C_0=\text{trivial},C_2=\text{standard},C_4=\text{standard}\otimes \text{sign},C_6=\text{sign},$ the graded lift of the regular representation, with graded multiplicities $1$ for the trivial module, $t^6$ for the sign module and $t^2+t^4$ for the two-dimensional standard module.

Character check. By The coinvariant algebra is the graded regular representation of W the graded character is $\mathrm{HTr}(C,t)(w)={\prod }_{i=1}^r(1-t^{2d_i})\det (1-t^{2}w\mid {\mathfrak{h}}^{\ast }{)}^{-1}\mathrm{.}$ For a transposition in $S_3$ the eigenvalues on ${\mathfrak{h}}^{\ast }$ are $1,-1$, so $\det (1-t^{2}w)=(1-t^2)(1+t^2)=1-t^4$ and the quotient is $(1-t^4)(1-t^6)\mathrm{/}(1-t^4)=1-t^6$, matching the direct sum $1+0\cdot t^2+0\cdot t^4+(-1)t^6$. For a three-cycle the eigenvalues are the primitive cube roots $\omega ,{\omega }^2$, so $\det (1-t^{2}w)=1+t^2+t^4$ and the quotient is $(1-t^4)(1-t^6)\mathrm{/}(1+t^2+t^4)=(1-t^2)(1-t^4)=1-t^2-t^4+t^6$, matching $1+(-1)t^2+(-1)t^4+t^6$. At $w=1$ the quotient is $1+2t^2+2t^4+t^6$ and its value at $t=1$ is $6=\mathrm{\mid }{S}_3\mathrm{\mid }$. In both cases the graded character at $t=1$ is the character of the regular representation, and the sum of graded multiplicities times dimensions is $1+2+2+1=6=\mathrm{\mid }W\mathrm{\mid }$, respectively $1+1=2=\mathrm{\mid }{S}_2\mathrm{\mid }$, as the general theorem asserts.

Assume the Axiom of Choice. Work in ${\mathfrak{s}\mathfrak{l}}_2$ with the conventions of The classical BGG category O: $\lambda =0$, $\rho =1$, so the regular standards are ${\mathrm{\Delta }}_1=M(0)$ and ${\mathrm{\Delta }}_s=M(-2)$ for the two elements of $W=\{1,s\}$, and the singular wall weight is $\mu =-1$, whose shifted value is $\mu +\rho =0$. The dominant regular integral block is that of $\lambda =0$ (The regular integral block and its antidominant weight), and the wall-crossing functor is that of The wall-crossing functor and its unit and counit built from the two-dimensional translating module $E=L(1)$ with highest weight $\mu -\lambda =-1$ up to Weyl conjugacy.

Onto the wall. Tensoring the two regular standards with the two-dimensional module and projecting to the central character of the wall preserves the unique wall standard: $T_{\mathrm{o}\mathrm{n}}{\mathrm{\Delta }}_1\cong M(-1)$ and $T_{\mathrm{o}\mathrm{n}}{\mathrm{\Delta }}_s\cong M(-1)$, since ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_s$ differ by the single simple-root lowering and the tensor filtration retains exactly the labels of weight $-1$ in the shifted parametrization, as in Translation to and from a single wall on standard modules.

Back out of the wall. The wall block at $\mu =-1$ has a single simple object $M(-1)=L(-1)$ and is semisimple; hence $M(-1)$ is projective and injective there, and the exact biadjoint return translation $T_{\mathrm{o}\mathrm{u}\mathrm{t}}$ sends it to a projective object of the regular block whose Verma flag has the two factors $M(0)$ and $M(-2)$, once each, by Translation to and from a single wall on standard modules. By biadjunction, its Hom to $L(-2)$ is one-dimensional, since $T_{\mathrm{o}\mathrm{n}}L(-2)=M(-1)$; its Hom to $L(0)$ is zero, since $L(1)\otimes L(0)=L(1)$ has a different central character from $M(-1)$. Thus its only simple quotient is $L(-2)$, once, and the projective-cover decomposition identifies it with $P(-2)$.

Conclusion. Both ${\theta }_s{\mathrm{\Delta }}_1$ and ${\theta }_s{\mathrm{\Delta }}_s$ are therefore isomorphic to the same nonsplit two-step extension $0⟶M(0)⟶P(-2)⟶M(-2)⟶0,$ with class $[{\mathrm{\Delta }}_1]+[{\mathrm{\Delta }}_s]$ in the Grothendieck group, in agreement with Wall-crossing acts on standard modules by a two-step extension. Maximal-weight peeling puts $M(0)$ first in the flag; the extension is nonsplit because $P(-2)$ is indecomposable as a projective cover. The wall sits at highest weight $-1$, equivalently at shifted parameter $\mu +\rho =0$; the regular highest weight $0$ is the dominant maximum of the block; the antidominant minimum is $-2$.

Assume the Axiom of Choice, inherited from the published items used. Work in type $A_2$: the Weyl group is $W=S_3=⟨s,t⟩$ with $s^2=t^2=1$ and $m_{st}=3$, so that the six elements $1,s,t,st,ts,sts$ are distinct, with $sts=tst=w_0$. Fix a dominant regular integral weight $\lambda$ and its regular integral block ${\mathcal{O}}_{\lambda }$ in the sense of The regular integral block and its antidominant weight, with standards $\mathrm{\Delta }(v\cdot \lambda )$ indexed by $v\in W$. By Simple and standard bases of K0(O) the classes $[\mathrm{\Delta }(v\cdot \lambda )]$ form a $\mathbb{Z}$-basis of $K_0({\mathcal{O}}_{\lambda })$, so the assignment $c\text{\hspace{0.17em}⁣}:K_0({\mathcal{O}}_{\lambda })⟶\mathbb{Z}[W],c[\mathrm{\Delta }(v\cdot \lambda )]=v$ is an isomorphism of abelian groups, and The wall-crossing action on the Grothendieck group is multiplication by 1+s identifies the class action of the wall-crossing functor ${\theta }_s$ of The wall-crossing functor and its unit and counit with right multiplication by $1+s$, and likewise for $t$.

The two-step products. Applying the class formula first to $\mathrm{\Delta }(\lambda )$ and then to the one-step object gives $c[{\theta }_t{\theta }_s\mathrm{\Delta }(\lambda )]=(1+s)(1+t)=1+s+t+st,$ because ${\theta }_s$ acts first and contributes the factor $1+s$ on the right. The other order gives $c[{\theta }_s{\theta }_t\mathrm{\Delta }(\lambda )]=(1+t)(1+s)=1+t+s+ts\mathrm{.}$ Thus $[{\theta }_t{\theta }_s\mathrm{\Delta }(\lambda )]=[\mathrm{\Delta }(\lambda )]+[\mathrm{\Delta }(s\cdot \lambda )]+[\mathrm{\Delta }(t\cdot \lambda )]+[\mathrm{\Delta }(st\cdot \lambda )]$ while $[{\theta }_s{\theta }_t\mathrm{\Delta }(\lambda )]=[\mathrm{\Delta }(\lambda )]+[\mathrm{\Delta }(t\cdot \lambda )]+[\mathrm{\Delta }(s\cdot \lambda )]+[\mathrm{\Delta }(ts\cdot \lambda )]\mathrm{.}$

Non-commutativity. The two products differ: the coefficients of the length-two elements $st$ and $ts$, which are distinct elements of $S_3$, are interchanged. Since the Verma classes are linearly independent, the two class operators on $K_0({\mathcal{O}}_{\lambda })$ differ, and in fact they already differ on the single class $[\mathrm{\Delta }(\lambda )]$. This is the same non-commutativity as in the Hecke algebra of the $S_3$ system, whose standard generators satisfy $T_s{T}_t=T_{st}\mathrm{\ne }{T}_{ts}=T_t{T}_s$; the wall-crossing functors accordingly behave Hecke-like rather than like a commuting tuple of group elements, and the two-step products are already sensitive to the order in which the walls are crossed.

Projective multiplicities. The object ${\theta }_s{\theta }_t\mathrm{\Delta }(\lambda )$ is projective, and its class contains the single Bruhat-maximal label $ts$ with coefficient one. By Wall-crossing compositions produce the projective with the top standard label for the reduced word $ts$ (the composite ${\theta }_s{\theta }_t$ applies ${\theta }_t$ first), the indecomposable projective $P(ts\cdot \lambda )$ occurs as a summand of ${\theta }_s{\theta }_t\mathrm{\Delta }(\lambda )$ with multiplicity one, and every other indecomposable summand has a strictly smaller label, i.e. lies among $P(1),P(s\cdot \lambda ),P(t\cdot \lambda )$. The multiplicity of $P(ts\cdot \lambda )$ is read off from the unitriangularity of the standard-support matrix of the block; the remaining multiplicities are determined recursively but the example records only the top one.

Assume the Axiom of Choice, inherited from the published items used. In type $A_2$ with distinct simple reflections $s,t$ (so $m_{st}=3$), the wall-crossing functors ${\theta }_s{\theta }_t$ and ${\theta }_t{\theta }_s$ on the regular integral block commute, so that the assignment $s\mapsto {\theta }_s$ defines a representation of $W=S_3$ on the block. This is false: the two functors are not naturally equivalent, and ${\theta }_s^2$ is not the identity.

Assume the inherited Axiom of Choice. In the regular integral ${\mathfrak{s}\mathfrak{l}}_2$ block at $\lambda =0$ (so $\rho =1$ and $s\cdot 0=-2$), the functor $V={\mathrm{Hom}}_{\mathcal{O}}(P(-2),-)$ is fully faithful on every object of the block, i.e. $V\text{\hspace{0.17em}⁣}:{\mathrm{Hom}}_{\mathcal{O}}(M,N)\to {\mathrm{Hom}}_{C\text{-mod}}(VM,VN)$ is a bijection for all objects $M,N$. This is false.

Assume the inherited Axiom of Choice. Work in the regular integral ${\mathfrak{s}\mathfrak{l}}_2$ block with $\lambda =0$, $\rho =1$, so $s\cdot 0=-2$ and the singular wall weight is $\mu =-1$ with $\mu +\rho =0$. The indecomposable projectives are $P_e=M(0)={\mathrm{\Delta }}_e$ and $P_s=P(-2)$, whose Verma flag is ${\mathrm{\Delta }}_e,{\mathrm{\Delta }}_s$ with each standard once (The regular integral block and its antidominant weight, Wall-crossing compositions produce the projective with the top standard label).

Set $\mathbb{k}=\mathbb{C}$.

On the standards. By Wall-crossing acts on standard modules by a two-step extension the wall-crossing functor ${\theta }_s$ of The wall-crossing functor and its unit and counit satisfies ${\theta }_s{\mathrm{\Delta }}_e={\theta }_s{\mathrm{\Delta }}_s=P_s$, the nonsplit two-step extension of ${\mathrm{\Delta }}_s$ by ${\mathrm{\Delta }}_e$: the one-letter reduced word $w=s$ gives ${\theta }_s\mathrm{\Delta }(\lambda )=P(s\cdot \lambda )$ as its unique new summand, and the flag computation ${\mathrm{\Delta }}_e+{\mathrm{\Delta }}_s$ shows the summand is the whole object. Moreover $T_{\mathrm{o}\mathrm{n}}{\mathrm{\Delta }}_e\cong T_{\mathrm{o}\mathrm{n}}{\mathrm{\Delta }}_s\cong M(-1)$ by the onto-wall standard formula in The wall-crossing functor and its unit and counit, so applying $T_{\mathrm{o}\mathrm{u}\mathrm{t}}$ gives ${\theta }_s{\mathrm{\Delta }}_s\cong {\theta }_s{\mathrm{\Delta }}_e=P_s$.

On the projectives. Exactness of ${\theta }_s$ (Translation functors are exact and biadjoint) applied to the flag $0\to {\mathrm{\Delta }}_e\to P_s\to {\mathrm{\Delta }}_s\to 0$ shows that ${\theta }_s{P}_s$ has standard flag multiplicities $({\mathrm{\Delta }}_e,{\mathrm{\Delta }}_s)=(2,2)$. Biadjointness of wall translation makes ${\theta }_s{P}_s$ projective. Every projective in this two-simple block is a direct sum of copies of $P_e$ and $P_s$, whose flag multiplicities are $(1,0)$ and $(1,1)$ by BGG reciprocity (BGG reciprocity); writing ${\theta }_s{P}_s\cong P_e^{\oplus a}\oplus P_s^{\oplus b}$, the equations $a+b=2$ and $b=2$ give $a=0$, $b=2$, so ${\theta }_s{P}_e\cong P_s,{\theta }_s{P}_s\cong P_s\oplus P_s\mathrm{.}$ In the ordered projective bases $(P_e,P_s)$, with columns indexed by the input, the matrix of ${\theta }_s$ is therefore $\left(\begin{array}{cc}0& 0\\ 1& 2\end{array}\right)$. On the Soergel side $C=\mathbb{k}[x]\mathrm{/}(x^2)$ with $\mathrm{deg}x=2$, $C^s=\mathbb{k}$, $V(P_e)=\mathbb{k}$ and $V(P_s)=C$; the wall-crossing theorem Wall-crossing under the Soergel functor is tensoring with the coinvariant bimodule gives $V{\theta }_s(P_e)=C{\otimes }_{C^s}\mathbb{k}\cong C,V{\theta }_s(P_s)=C{\otimes }_{C^s}C\cong C\oplus C$ as left $C$-modules, the left $C$-acting on the first factor, and the projective Struktursatz Soergel's Struktursatz: the functor V is fully faithful on projectives reflects these isomorphisms back to the projective identities above, all objects involved being projective.

Dimension comparison. The tensor $C{\otimes }_{C^s}C\cong C{\otimes }_{\mathbb{k}}C$ is the four-dimensional coinvariant bimodule $=2\mathrm{\mid }W\mathrm{\mid }$ with $\mathrm{\mid }W\mathrm{\mid }=2$, explicitly spanned by the four elements $1\otimes 1,1\otimes x,x\otimes 1,x\otimes x$; the isomorphism $C\oplus C\to C{\otimes }_{\mathbb{k}}C$ of left $C$-modules sends $(u,v)$ to $u\otimes 1+v\otimes x$. The polynomial bimodule $B_s=\mathbb{k}[a]{\otimes }_{\mathbb{k}[a^2]}\mathbb{k}[a](1)$ is free of rank two over the right $\mathbb{k}[a]$, and is infinite-dimensional over $\mathbb{k}$. Its right origin reduction ${\bar{B}}_s=B_s{\otimes }_{\mathbb{k}[a]}\mathbb{k}\cong C(1)$ is two-dimensional, with degrees $-1,1$ after centering, whereas the coinvariant bimodule $C{\otimes }_{\mathbb{k}}C$ has dimension four. The former models $V{\theta }_s(P_e)$ and the latter the tensor operator applied to $V(P_s)=C$.

Assume the Axiom of Choice inherited from the category-$\mathcal{O}$ suppliers. Work in the dominant regular integral ${\mathfrak{s}\mathfrak{l}}_2$ block at $\lambda =0$, with $\rho =1$, $W=\{e,s\}$ and $s\cdot 0=-2$ (The regular integral block and its antidominant weight). Set $\mathbb{k}=\mathbb{C}$. Put ${\mathrm{\Delta }}_e=M(0)$, ${\mathrm{\Delta }}_s=M(-2)=L(-2)$, $P_e=P(0)={\mathrm{\Delta }}_e$ and $P_s=P(-2)$; the block is described by the nonsplit sequences $0\to L(-2)\to {\mathrm{\Delta }}_e\to L(0)\to 0,0\to {\mathrm{\Delta }}_e\to P_s\to {\mathrm{\Delta }}_s\to 0,$ the second being the one-letter case of Wall-crossing acts on standard modules by a two-step extension and Wall-crossing compositions produce the projective with the top standard label, and $P_e={\mathrm{\Delta }}_e$ being the dominant Verma (BGG reciprocity, Verma modules).

The values of $V$. Let $C=\mathbb{k}[x]\mathrm{/}(x^2)$ with $\mathrm{deg}x=2$ be the coinvariant algebra and $V=\mathrm{Hom}(P_s,-)$ (The antidominant projective and the Soergel functor), so that $\mathrm{End}(P_s)\cong C$ by The endomorphisms of the antidominant projective form the coinvariant algebra. Since $P_s$ has indecomposable head $L(-2)$ and $L(-2)=M(-2)$ is simple, the left $C$-module values are $V(P_e)=\mathbb{k}=C\mathrm{/}(x),V(P_s)=C,V({\mathrm{\Delta }}_e)=V({\mathrm{\Delta }}_s)=\mathbb{k},V(L(0))=0,V(L(-2))=\mathbb{k}\mathrm{.}$ Indeed $V({\mathrm{\Delta }}_e)$ and $V({\mathrm{\Delta }}_s)$ are one-dimensional by exactness applied to the first sequence and $V(L(-2))=\mathbb{k}$ from the head quotient, while $V(L(0))=0$ because $L(0)$ is not a quotient of $P_s$; the unique one-dimensional $C$-module is $\mathbb{k}$ with $x$ acting by zero.

The projective Hom table. With rows indexed by the source and columns by the target, the four Hom spaces in the ordered bases $(P_e,P_s)$ are $\left(\begin{array}{cc}\mathrm{Hom}(P_e,P_e)& \mathrm{Hom}(P_e,P_s)\\ \mathrm{Hom}(P_s,P_e)& \mathrm{Hom}(P_s,P_s)\end{array}\right)=\left(\begin{array}{cc}\mathbb{k}& \mathbb{k}\\ \mathbb{k}& C\end{array}\right),\text{dimensions }\left(\begin{array}{cc}1& 1\\ 1& 2\end{array}\right)\mathrm{.}$ Choose the nonzero maps $i\text{\hspace{0.17em}⁣}:P_e\to P_s$ and $p\text{\hspace{0.17em}⁣}:P_s\to P_e$; under the Struktursatz Soergel's Struktursatz: the functor V is fully faithful on projectives their images in ${\mathrm{Hom}}_C(\mathbb{k},C)=\mathbb{k}{i}_0$ and ${\mathrm{Hom}}_C(C,\mathbb{k})=\mathbb{k}{p}_0$ are the generators $i_0(1)=x$ and $p_0(a+bx)=a$, and the compositions compute $p_0{i}_0=0$ on $\mathbb{k}$ while $i_0{p}_0=m_x$, multiplication by $x$ on $C$ (equivalently $pi=0$ and $ip$ is a nonzero square-zero endomorphism; rank one refers to its image under $V$ on $C$). In particular $\mathrm{End}(P_s)\cong C$ is recovered as the full two-dimensional endomorphism algebra, and the full-faithfulness comparison is computed on the four projective Hom spaces rather than claimed for the whole block.

Polynomial lifts and standard ranks. For $S=\mathbb{k}[a]$ with $s(a)=-a$, the empty-word lift is $B_e=S$ and the rank-one lift is $B_s=S{\otimes }_{\mathbb{k}[a^2]}S(1)$ (The Soergel bimodule $B_i$ of a simple reflection); their right origin reductions are $\mathbb{k}$ and $C(1)$, whose centered auxiliary degrees are $-1,1$, and the labels $e^{-1}=e$, $s^{-1}=s$ agree with the dictionary of Indecomposable projectives correspond to indecomposable Soergel modules. Before reduction the standard graph ranks, in the order $(e,s)$, are $(1,0)$ for $B_e$ and $(1,1)$ for $B_s$: in the two-variable model $B_s=\mathbb{k}[l,r]\mathrm{/}(l^2-r^2)(1)$ the submodule generated by $l-r$ is free of rank one on the graph $l=-r$ with annihilator $(l+r)$, and the quotient is free of rank one on the graph $l=r$, with annihilator $(l-r)$ (Standard graph bimodules, support filtrations and characters). These ranks agree with the Verma-flag multiplicities $(1,0)$ of $P_e$ and $(1,1)$ of $P_s$ at $v=1$, as predicted by Verma-flag multiplicities are standard-filtration ranks under V; they are taken before the graph supports coalesce at the origin, and no grading on arbitrary category-$\mathcal{O}$ objects or intrinsic separated-support filtration on $V(P)$ is asserted.