Singular and Parabolic Kazhdan–Lusztig Theory — Examples

The ordinary values needed. By [F2] the comparable pairs needed below are $(e,e)$ (gap 0), $(e,s_2)$ (gap 1), $(s_2,s_2)$, $(s_1,s_1)$, $(s_1,s_1{s}_2)$ (gap 1), $(s_1{s}_2,s_1{s}_2)$, $(e,s_2{s}_1)$ (gap 2), $(s_1,s_2{s}_1)$ (gap 1), $(s_2,s_2{s}_1)$ (gap 1), $(s_2{s}_1,s_2{s}_1)$, $(s_1,w_0)$ (gap 2), $(s_1{s}_2,w_0)$ (gap 1), $(s_2,w_0)$ (gap 2), $(w_0,w_0)$; by [F1] the values are $1$ on the diagonal, $v$ on the listed gap-one pairs, $v^2$ on the listed gap-two pairs, and zero on non-comparable pairs such as $(s_1,s_2)$, $(s_1{s}_2,s_2{s}_1)$, $(s_1{s}_2,s_2)$, $(s_1,w_0)$'s reverse orders, and all pairs $y>x$.

Antispherical table. Using [F3] and step 1.1, $W_J=\{1,s_1\}$ contributes the two terms $z=e$ and $z=s_1$: $n_{e,e}=p_{e,e}=1$, $n_{e,s_2}=p_{e,s_2}-v{p}_{s_1,s_2}=v-0=v$, $n_{e,s_2{s}_1}=p_{e,s_2{s}_1}-v{p}_{s_1,s_2{s}_1}=v^2-v^2=0$; $n_{s_2,s_2}=p_{s_2,s_2}-v{p}_{s_1{s}_2,s_2}=1-0=1$ (the second pair is not comparable since $\mathrm{\ell }(s_1{s}_2)>\mathrm{\ell }(s_2)$), $n_{s_2,s_2{s}_1}=p_{s_2,s_2{s}_1}-v{p}_{s_1{s}_2,s_2{s}_1}=v-0=v$ (the pair $s_1{s}_2,s_2{s}_1$ is incomparable); $n_{s_2{s}_1,s_2{s}_1}=p_{s_2{s}_1,s_2{s}_1}-v{p}_{w_0,s_2{s}_1}=1-0=1$ (the pair $w_0,s_2{s}_1$ has $y>x$). This is the full antispherical table of the Example, and coefficients with $y\le ̸x$ are zero by the support clause of the basis theorem.

Spherical table. Using [F4] with $w_0^J=s_1$ and step 1.1: $m_{e,e}=p_{s_1,s_1}=1$, $m_{e,s_2}=p_{s_1,s_1{s}_2}=v$, $m_{e,s_2{s}_1}=p_{s_1,w_0}=v^2$; $m_{s_2,s_2}=p_{s_1{s}_2,s_1{s}_2}=1$, $m_{s_2,s_2{s}_1}=p_{s_1{s}_2,w_0}=v$; $m_{s_2{s}_1,s_2{s}_1}=p_{w_0,w_0}=1$. This is the full spherical table of the Example.

Comparison at $v=1$ and conclusion. Setting $v=1$ in steps 2.1 and 2.2 gives the two displayed integer unitriangular matrices; they differ exactly in the entry $(e,s_2{s}_1)$, where the antispherical value is $0$ and the spherical value is $1=v^2{\mathrm{\mid }}_{v=1}$. Hence the two normalizations already differ for the smallest nontrivial parabolic, and neither table can be substituted for the other.

For $y=e$ and $x=s_2{s}_1$ the antispherical alternating sum [F3] has the two terms $z=e$ and $z=s_1$: $n_{e,s_2{s}_1}=p_{e,s_2{s}_1}-v{p}_{s_1,s_2{s}_1}$. By [F1] the first value has length gap two, so $p_{e,s_2{s}_1}=v^2$; the second has gap one (by [F2], $s_1<s_2{s}_1$), so $p_{s_1,s_2{s}_1}=v$. Hence $n_{e,s_2{s}_1}=v^2-v\cdot v=0$.

For the same pair the spherical reduction [F4] gives $m_{e,s_2{s}_1}=p_{s_{1}e,s_1{s}_2{s}_1}=p_{s_1,w_0}$; the pair has length gap two by [F2] ($\mathrm{\ell }(w_0)-\mathrm{\ell }(s_1)=3-1=2$), so by [F1] $p_{s_1,w_0}=v^2$ and $m_{e,s_2{s}_1}=v^2$.

Specializing at $v=1$ gives $n_{e,s_2{s}_1}(1)=0\mathrm{\ne }1=m_{e,s_2{s}_1}(1)$: no unit of $\mathbb{Z}$ (indeed no unit of $\mathbb{C}$) multiplied by $0$ gives $1$, so the antispherical value is not a unit multiple of the spherical one; and since the two elements $0$ and $v^2$ of $\mathbb{Z}[v]$ are distinct, they differ as polynomials and not only after specialization.

Consequently the two normalizations cannot be substituted for one another in a multiplicity statement: a singular-block or parabolic multiplicity theorem must fix the side of the parabolic and the corresponding module. In particular the singular-block multiplicity theorem of this page is stated in the spherical normalization, whereas a genuine regular parabolic locally finite category uses the antispherical family; replacing the spherical value $m_{e,s_2{s}_1}(1)=1$ by the antispherical value $n_{e,s_2{s}_1}(1)=0$ produces the false multiplicity $0$ for this pair. The computation refutes the normalization-blind claim, not the parabolic basis theorem or the alternating-sum identity that produced the two values.

The sl_2 singular block. For $\mu =-1=-\rho$ the dot orbit is $\{-1\}$, so ${\mathcal{O}}_{-1}$ has the single simple $L(-1)$, and by [F2] $M(-1)=L(-1)=P(-1)=\mathrm{\nabla }(-1)$ is projective and injective.

The sl_2 translation. Here $W^J=\{1\}$. By [F1], $TM(-2)=M(-1)$ and $TM(0)=M(-1)$; and on simples $TL(-2)=L(-1)$, while $TL(0)=0$ because $s\mathrm{\notin }{W}^J$.

The sl_3 matrices. For $y=(1,s_2,s_1{s}_2)$ the products $y{w}_0=(w_0,s_1{s}_2,s_2)$ form the descending Bruhat chain $w_0>s_1{s}_2>s_2$ by reduced subwords. Their supported length gaps are zero, one or two. Thus [F3] gives the displayed LOWER unitriangular matrix $A$: its row $y$, column $w$ is one exactly when $y\ge w$. BGG [F4] gives the same entries for $(P(w\cdot \mu ):\mathrm{\Delta }(y\cdot \mu ))$. Thus the column at $w=1$ consists of three ones, the column at $w=s_2$ has zeros at $y=1$ and ones at $y=s_2,s_1{s}_2$, and the column at $w=s_1{s}_2$ has only the diagonal one. These are precisely the corrected projective flags displayed above; reading rows gives the unchanged simple composition factors of the three Vermas.

The sl_2 projective. The relations in [F4] give $e{f}^k{v}_l=k(l-k+1)f^{k-1}{v}_l$ by induction using $[e,f]=h$ and $[h,f]=-2f$. For $l=0$, the span of $f^k{v}_0$, $k\ge 1$, is the copy of $M(-2)$ generated by $f{v}_0$, and the quotient is the one-dimensional $L(0)$. Since $M(-2)$ is simple, BGG [F4] gives $(P(-2):\mathrm{\Delta }(-2))=(P(-2):\mathrm{\Delta }(0))=1$; the rank-one dot orbit has no further labels. Applying exact $T$ to this Verma flag gives two factors $\mathrm{\Delta }(-1)=P(-1)$ by [F1] and step 1.1. Every extension of these projectives splits, so $TP(-2)\cong P(-1)\oplus P(-1)$.

Surviving labels. Applying [F1] to the simple labels: $L(w\cdot \lambda )$ survives translation iff $w\in W^J=\{1,s_2,s_1{s}_2\}$; hence $L(s_1\cdot \lambda )$ and $L(w_0\cdot \lambda )=L(s_1{s}_2{s}_1\cdot \lambda )$ are sent to zero, as are all other non-minimal labels. This proves every displayed assertion.

The parabolic matrix. By [F2] the matrix $N=(n_{y,x}(1))$ in the order $(e,s_2,s_2{s}_1)$ is the displayed upper unitriangular matrix. The displayed matrix $N^{-1}$ is its inverse: $N{N}^{-1}$ has entries ${\sum }_z{n}_{y,z}(1)(N^{-1}{)}_{z,x}$, which equal $1$ for $y=x$ and $0$ otherwise (the only nonzero contributions are from $z\in \{y,\dots ,x\}$ with the alternating coefficients), and similarly $N^{-1}N=I$. Both products are finite integer calculations.

The singular matrix. By [F3] the composition matrix in the order $(e,s_2,s_1{s}_2)$ is $A=\left(\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 1& 1\end{array}\right)$ (entry $(y,w)$ is $1$ exactly when $y\ge w$). The displayed matrix $B=\left(\begin{array}{ccc}1& 0& 0\\ -1& 1& 0\\ 0& -1& 1\end{array}\right)$ satisfies $AB=I=BA$ by the same finite chain computation, so $A^{-1}=B$.

The parabolic characters. By [F1] and step 1.1, $\mathrm{ch}L(x\cdot \lambda )={\sum }_y(N^{-1}{)}_{x,y}\mathrm{ch}{\mathrm{\Delta }}^J(y\cdot \lambda )$. Reading the rows of $N^{-1}$ in the order $(e,s_2,s_2{s}_1)$ gives exactly the three displayed character identities, using $0$, $-{\alpha }_2$, $-{\alpha }_1-2{\alpha }_2$ as the labels of $e\cdot 0$, $s_2\cdot 0$, $s_2{s}_1\cdot 0$ by [F5]: with $\rho ={\alpha }_1+{\alpha }_2$, the type-$A_2$ reflection relations give $s_2\rho ={\alpha }_1$ and $s_2{s}_1\rho =-{\alpha }_2$, hence the stated dot translates after subtracting $\rho$.

The singular characters and the comparison. By [F3] and [F4], $\mathrm{ch}L(w\cdot \mu )={\sum }_y(A^{-1}{)}_{w,y}\mathrm{ch}M(y\cdot \mu )$; the rows of $B=A^{-1}$ give the three displayed singular identities, in which the nonzero off-diagonal coefficients are the Mobius values $\pm 1$ of this all-one chain only. The parabolic coefficients of step 2.1 include the value $+1$ in the $(1,3)$ position of $N^{-1}$, which the singular inverse does not have: the two formulas are different character formulas for different blocks, and the singular one is not the generalized-Verma character formula of the parabolic category.