Primitive subspaces, hard Lefschetz and the Hodge-Riemann signs: the dictionary of hypotheses S, hL and HR

The linear-algebraic data. Let $H={⨁}_{i\in \mathbb{Z}}{H}^i$ be a finite-dimensional graded real vector space, concentrated in a single parity (all nonzero $H^i$ have $i\equiv j(\mathrm{m}\mathrm{o}\mathrm{d}2)$ for one $j\in \{0,1\}$), equipped with a graded symmetric bilinear form $⟨-,-⟩$: symmetric in the sense of Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms and graded in the sense $⟨a,b⟩=0$ unless $\mathrm{deg}a+\mathrm{deg}b=0$, non-degenerate in the sense that the induced map $H\to H^{\ast }$ is an isomorphism; definiteness, inertia and signature are used in the sense of Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form, and Sylvester's law Sylvester's law of inertia: every real symmetric form is congruent to $\mathrm{diag}(I_p,-I_q,0_r)$, and $(p,q,r)$ is unique supplies the uniqueness of the inertia. Let $L:H\to H$ be a symmetric degree-two operator, i.e. $L(H^i)\subseteq H^{i+2}$ and $⟨La,b⟩=⟨a,Lb⟩$ for all $a,b$.

Hard Lefschetz. The operator $L$ satisfies hard Lefschetz (hL) when $L^i:H^{-i}⟶H^i$ is an isomorphism for every $i\ge 0$. Define the primitive subspaces $P_L^{-i}:=\ker \left(L^{i+1}{\mathrm{\mid }}_{H^{-i}}\right)\subseteq H^{-i}(i\ge 0),$ and the Lefschetz forms $(a,b{)}_L^{-i}:=⟨a,L^{i}b⟩(a,b\in H^{-i},i\ge 0)\mathrm{.}$ Under hL the primitive decomposition $H={⨁}_{i\ge 0}{⨁}_{j=0}^i{L}^j{P}_L^{-i}$ holds; it is proved in the primitive-decomposition lemma later on this page, together with the non-degeneracy of the Lefschetz forms and the signature formula for them.

The Hodge-Riemann sign. Assume $H\mathrm{\ne }0$ and let $-d$ be its lowest nonzero degree, so $H^{-d}\mathrm{\ne }0$ and $H^i=0$ for $i<-d$. The standard sign at degree $-i$ is $(-1{)}^{(d-i)\mathrm{/}2}$; when $d-i$ is odd the primitive space $P_L^{-i}$ is declared zero and the sign is not used. The pair $(H,L,⟨-,-⟩)$ satisfies the Hodge-Riemann bilinear relations with the standard sign (HR) when hL holds and the Lefschetz form $(a,b{)}_L^{-i}$ on $P_L^{-i}$ is definite of sign $(-1{)}^{(d-i)\mathrm{/}2}$ for every $i\ge 0$, with the bottom space $P_L^{-d}=H^{-d}$ positive definite. HR with global sign $\epsilon \in \{\pm 1\}$ means that the same statement holds with $(-1{)}^{(d-i)\mathrm{/}2}$ replaced by $\epsilon (-1{)}^{(d-i)\mathrm{/}2}$; the standard sign is thus the normalization that makes the bottom form positive.

The hypotheses S, hL, HR for Soergel bimodules. For $x\in S_n$ let $B_x^R$ be the indecomposable real Soergel bimodule with its reduced module $\overline{B_x^R}$ (The reduced Soergel module and the dominant Lefschetz operator). The statement $\mathrm{S}(x):\mathrm{ch}(B_x^R)={\underset{‾}{H}}_x$ means that the character of $B_x^R$ is the Kazhdan-Lusztig basis element ${\underset{‾}{H}}_x$ in the Hecke algebra of the type-A page. On $\overline{B_x^R}$ use the centered grading, the operator $L:=\rho \cdot (-)$ of left multiplication by a strictly dominant $\rho$ as in The reduced Soergel module and the dominant Lefschetz operator, and the positively normalized intrinsic invariant form when $\mathrm{S}(x)$ holds: the unique-up-to-scalar non-degenerate invariant form of The trace, the invariant intersection form on Bott-Samelson bimodules and its reduction, normalized so that the bottom Lefschetz form $⟨{\rho }^{\mathrm{\ell }(x)}{c}_{\mathrm{b}\mathrm{o}\mathrm{t}},c_{\mathrm{b}\mathrm{o}\mathrm{t}}⟩$ on the one-dimensional space ${\overline{B_x^R}}^{-\mathrm{\ell }(x)}$ equals $+1$. Here $d=\mathrm{\ell }(x)$ by the proved normalized degree bounds. The property $\mathrm{h}\mathrm{L}(x)$ always denotes hard Lefschetz for this operator on $\overline{B_x^R}$, independently of $\mathrm{S}(x)$. $\mathrm{H}\mathrm{R}(x)$ denotes the conjunction of $\mathrm{S}(x)$ and the Hodge-Riemann relations with the standard sign for this normalized intrinsic form; its existence and uniqueness, and the nonzero positive bottom value used for normalization, are proved in Invariant forms on an indecomposable Soergel bimodule are unique up to scalar, non-degenerate, and agree with the restricted Bott-Samelson form ↗. The embedded-form statements refer to a reduced expression $\underset{‾}{x}$ of $x$ and an embedding $B_x^R\subseteq \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}⨁\mathrm{B}\mathrm{S}(\underset{‾}{x})$ as a direct summand: then $\mathrm{H}\mathrm{R}\mathrm{b}\mathrm{a}\mathrm{r}(x)$ requires the general Hodge-Riemann relations above for the restricted intersection form through every such embedding, and its equivalence with $\mathrm{H}\mathrm{R}(x)$ (the comparison scalar is positive) is the content of the invariant-form uniqueness lemma later on this page. The strong embedded-form statement $\mathrm{H}\mathrm{R}\mathrm{b}\mathrm{a}\mathrm{r}(x)$ is formulated for the restricted forms and makes sense without assuming $\mathrm{S}(x)$.

The ascent and deformation packages. Fix $xs>x$ and put $M=B_x^R{B}_s^R$, whose reduced module has lowest degree $-\mathrm{\ell }(x)-1$; the ascent packages are read with $d=\mathrm{\ell }(x)+1,L=L_0\text{ or }{L}_{\zeta },$ and inherit hL, HR, HRbar, and the sign $(-1{)}^{(\mathrm{\ell }(x)+1-i)\mathrm{/}2}$ from the general definition above, where $L_{\zeta }$ is the deformed Lefschetz operator defined later on this page. Thus $\mathrm{h}\mathrm{L}(x,s)$, $\mathrm{H}\mathrm{R}(x,s)$ refer to a fixed reduced-word induced form on $M$, while $\mathrm{H}\mathrm{R}\mathrm{b}\mathrm{a}\mathrm{r}(x,s)$ requires the statement for every choice of reduced-word embeddings and induced forms; the subscript $\zeta$ denotes the same statements for $L_{\zeta }$. The local sign hypotheses ${\mathrm{S}}^{\pm }(y,x,s)$ concern the local intersection form $(-,-{)}_y^{x,s}$ on ${\mathrm{Hom}}^0(B_y,M)$ defined later on this page: ${\mathrm{S}}^{+}(y,x,s)$ means $(-1{)}^{(\mathrm{\ell }(x)+1-\mathrm{\ell }(y))\mathrm{/}2}$-definiteness of that form and ${\mathrm{S}}^{-}(y,x,s)$ the opposite definiteness, with the parity convention of the primitive subspaces.

Orthogonal sums. If $H={⨁}_{\alpha }{H}_{\alpha }$ is an orthogonal direct sum of $L$-stable graded subspaces and $H$ has lowest degree $-d$, then HR holds for $(H,L,⟨-,-⟩)$ if and only if each summand satisfies hL and its primitive Lefschetz forms are definite with the ambient signs $(-1{)}^{(d-i)\mathrm{/}2}$; these need not be the summand's own standard signs when the summands have different lowest degrees. The sign bookkeeping for orthogonal blocks is used in the tensoring lemma of this page.

• Parity form of the sign. If $H$ is concentrated in parity $j$ and $i\equiv j(\mathrm{m}\mathrm{o}\mathrm{d}2)$, then $(-1{)}^{(d-i)\mathrm{/}2}=(-1{)}^{(d-j)\mathrm{/}2}(-1{)}^{(i-j)\mathrm{/}2}$; the constant factor $(-1{)}^{(d-j)\mathrm{/}2}$ is absorbed into the global sign $\epsilon$, so "HR with the standard sign" is equivalent to "hL plus $\epsilon$-definiteness with $\epsilon$ chosen to make the bottom form positive", and no sign is imposed twice.
• Half-integer exponents. When $d-i$ is odd, $(-1{)}^{(d-i)\mathrm{/}2}$ is not an integer power of $-1$; the convention is that the corresponding primitive space is zero. This is consistent with the parity concentration: primitive spaces occur only in degrees $\equiv d(\mathrm{m}\mathrm{o}\mathrm{d}2)$, and degrees of the opposite parity are zero in $H$.
• Non-degenerate versus definite. Non-degeneracy of the form on $H$ does not imply any definiteness of the Lefschetz forms; HR is an additional hypothesis, and in the tensoring lemma the definiteness of a multiplicity-space pairing is an input, not a consequence of non-degeneracy.
• The zero space. If $H=0$ there is no bottom degree; all statements hold vacuously, and this case is recorded as inapplicable in every contract of this page that quantifies over a nontrivial module.