Short Star-Products for Filtered Quantizations, I

Short Star-Products for Filtered Quantizations, I††This paper is a
contribution to the Special Issue on Algebra, Topology, and Dynamics in
Interaction in honor of Dmitry Fuchs. The full collection is available at
https://www.emis.de/journals/SIGMA/Fuchs.html

Pavel ETINGOF and Douglas STRYKER P. Etingof and D. Stryker Department of
Mathematics, MIT, Cambridge, MA 02139, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received October 01, 2019, in final form March 01, 2020; Published online
March 11, 2020

We develop a theory of short star-products for filtered quantizations of
graded Poisson algebras, introduced in 2016 by Beem, Peelaers and Rastelli for
algebras of regular functions on hyperKähler cones in the context of
3-dimensional $N=4$ superconformal field theories [Beem C., Peelaers W.,
Rastelli L., Comm. Math. Phys. 354 (2017), 345–392]. This appears to be a new
structure in representation theory, which is an algebraic incarnation of the
non-holomorphic ${\rm SU}(2)$-symmetry of such cones. Using the technique of
twisted traces on quantizations (an idea due to Kontsevich), we prove the
conjecture by Beem, Peelaers and Rastelli that short star-products depend on
finitely many parameters (under a natural nondegeneracy condition), and also
construct these star products in a number of examples, confirming another
conjecture by Beem, Peelaers and Rastelli.

star-product; quantization; hyperKähler cone; symplectic singularity

06B15; 53D55

To Dmitry Borisovich Fuchs on his 80th birthday with admiration

##  1 Introduction

The goal of this paper is to develop a theory of short star-products for
filtered quantizations of graded Poisson algebras, introduced in 2016 by Beem,
Peelaers and Rastelli for algebras of regular functions on hyperKähler cones
in the context of 3-dimensional $N=4$ superconformal field theories [1].

Namely, let $A$ be a commutative ${\mathbb{Z}}_{\geq 0}$-graded
${\mathbb{C}}$-algebra with a Poisson bracket $\\{\,,\,\\}$ of degree $-2$,
and let $*$ be a star-product on $A$ quantizing $\\{\,,\,\\}$. This means that
$*$ is an associative product on $A$ defined by

$\displaystyle a*b=ab+C_{1}(a,b)+C_{2}(a,b)+\cdots,\qquad a,b\in A,$

where $C_{k}\colon A\otimes A\to A$ are bilinear maps of degree $-2k$ such
that $C_{1}(a,b)-C_{1}(b,a)=\\{a,b\\}$.111Thus, we set the Planck constant
$\hbar$ to be $1$, which can be done without loss of generality because of the
grading on $A$. For degree reasons, for any homogeneous $a,b\in A$, one has
$C_{k}(a,b)=0$ for $k>\frac{\deg(a)+\deg(b)}{2}$, so the sum makes sense.

Star-products arise naturally from filtered quantizations of $A$. Namely, let
$d\colon A\to A$ be the degree operator, i.e., $d(a)=ma$ for $a\in A_{m}$, and
let $s=(-1)^{d}\colon A\to A$ be the operator such that $s(a)=(-1)^{m}a$ for
$a\in A_{m}$.222As customary in physical literature, we will abuse notation by
writing $d$ both for the degree of an element of $A$ and the degree operator
$A\to A$ whose eigenvalues are such degrees. Then $s$ is a Poisson
automorphism of $A$ defining an action of ${\mathbb{Z}}/2$ on $A$. Let
${{\mathbf{A}}}$ be a ${\mathbb{Z}}/2$-equivariant filtered quantization of
$A$, and let $\phi\colon A\to{{\mathbf{A}}}$ be a ${\mathbb{Z}}/2$-equivariant
quantization map, i.e., a ${\mathbb{Z}}/2$-equivariant filtration-preserving
linear map such that $\operatorname{gr}\phi={\rm id}$ (it attaches to a
“classical observable” $a\in A$ the corresponding “quantum observable”
$\widehat{a}:=\phi(a)$). Then the formula

$\displaystyle a*b:=\phi^{-1}(\phi(a)\phi(b))$

defines a star-product on $A$.

Conversely, any star-product on $A$ defines a filtered quantization of $A$
realized as a new product on the same vector space, and this is a traditional
method of quantizing Poisson algebras. For example, the well known Moyal–Weyl
star-product (Example 2.8 below) gives a quantization of the standard
$2n$-dimensional classical phase space which goes back to the early years of
quantum mechanics.

This classical story has recently had an exciting new development. Namely,
motivated by 3-dimensional N=4 superconformal field theories, in 2016 Beem,
Peelaers and Rastelli [1] considered a special class of star-products which we
call short, i.e., such that for any homogeneous $a,b\in A$ one has
$C_{k}(a,b)=0$ for any $k>\min(\deg(a),\deg(b))$. In other words, the degree
of any term in $a*b$ is at least $|\deg(a)-\deg(b)|$ (in [1] this is called
the truncation condition). Beem, Peelaers and Rastelli showed how short star-
products appear naturally in 3-dimensional N=4 superconformal field theories,
defining filtered quantizations of the corresponding Higgs and Coulomb
branches $X$, and conjectured that any even filtered quantization of
$A={\mathcal{O}}(X)$ (i.e., a ${\mathbb{Z}}/2$-equivariant quantization
${{\mathbf{A}}}$ equipped with a filtered antiautomorphism
$\sigma\colon{{\mathbf{A}}}\to{{\mathbf{A}}}$ such that
$\operatorname{gr}(\sigma)={\rm i}^{d}$ and $\sigma^{2}=s$) admits an even
short star-product (i.e., such that $C_{k}(a,b)=(-1)^{k}C_{k}(b,a)$).
Moreover, they predicted that such star-products depend on finitely many
parameters. Finally, they conjectured the existence of an even short star-
product satisfying a positivity condition; namely, this condition should hold
for the star-product arising from the physical theory.

There are several examples (given in [1]) in which this conjecture is easy to
prove, as the even short star-product is uniquely determined by the symmetry
of the situation. For instance, if $V$ is a finite dimensional symplectic
vector space and $A=\mathcal{O}(V)$ then the symmetric Moyal–Weyl star-product
on $A$ is short and even. Since this star-product is ${\rm Sp}(V)$-invariant,
for any finite subgroup $G\subset{\rm Sp}(V)$ this gives an even short star-
product on ${\mathcal{O}}(V/G)=\mathcal{O}(V)^{G}$. Also, if
$X\subset\mathfrak{g}^{*}$ is the closure of the minimal nilpotent orbit of a
simple Lie algebra $\mathfrak{g}$, then any filtered (even) quantization of
$\mathcal{O}(X)$ (with degrees multiplied by $2$) may be defined by a
$\mathfrak{g}$-invariant (even) star-product (see [22]), which is short for
representation-theoretic reasons.333The quantization of the minimal orbit is
the quotient of ${\rm U}(\mathfrak{g})$ by the so-called Joseph ideal. This
quantization (and ideal) is unique if $\mathfrak{g}\neq\mathfrak{sl}_{n}$ and
depends on one parameter $\lambda$ for $\mathfrak{g}=\mathfrak{sl}_{n}$. The
even quantization is unique except $\mathfrak{g}=\mathfrak{sl}_{2}$, where all
quantizations are even. E.g., for $\mathfrak{g}=\mathfrak{sl}_{n}$ with $n\geq
3$, the quantization is even if and only if $\lambda=0$.

In particular, this applies to the $A_{1}$-singularity, which corresponds to
the case of $\mathfrak{g}=\mathfrak{sl}_{2}$. Namely, in this case $X$ is the
usual quadratic cone in ${\mathbb{C}}^{3}$, so $\mathcal{O}(X)=\oplus_{m\in
2{\mathbb{Z}}_{\geq 0}}V_{m}$, where $V_{m}$ is the irreducible representation
of ${\rm SU}(2)$ with highest weight $m$, and the shortness property comes
from the fact that if $V_{k}$ is contained in $V_{m}\otimes V_{n}$ then
$k\geq|m-n|$ (the Clebsch–Gordan rule). In fact, as explained in [1], the same
mechanism, but for the non-holomorphic ${\rm SU}(2)$-symmetry of the
hyperKähler cone acting by rotations on its 2-sphere of complex structures, is
responsible for appearance of short star-products in 3-dimensional $N=4$
superconformal field theories.

However, the shortness condition for a star-product is a very strong
restriction, and from the purely mathematical viewpoint the existence of short
star-products for a given quantization is far from obvious. In general it is
not even clear if there exists any short star-product at all. In fact, the
existence of a short star-product is already non-obvious for quantizations of
Kleinian singularities beyond the $A_{1}$ case, although some low-degree
calculations of [1] for $A_{2}$ and $A_{3}$ showed that the existence of
positive even short star-products (for even quantizations) is highly
plausible, and this was confirmed later in [7, 9] for $A_{n}$ for all $n$.
Also, examples of short star-products for $D_{n}$ singularities are
constructed in [8, Section 5.2.1].

The goal of this paper is to partially prove the conjectures from [1]. Namely,
let us say that a short star-product $*$ is nondegenerate if the bilinear form
$(a,b)={\rm CT}(a*b)$ on $A$ (where ${\rm CT}$ stands for the constant term)
is nondegenerate. This is a natural condition from the physics point of view;
e.g., it follows from the positivity condition, saying that the Hermitian
version of this form is positive definite. Our first main result is that
nondegenerate short star-products for hyperKähler cones depend on finitely
many parameters, confirming a conjecture from [1]. The proof is based on the
idea of Kontsevich to express nondegenerate short star-products in terms of
nondegenerate twisted traces, which can then be understood as elements of a
certain zeroth Hochschild homology group, as well as a number of results on
symplectic singularities [17, 19, 25], hyperKähler manifolds [16] and Poisson
homology [11].

Our second main result is the existence and classification of nondegenerate
short star-products in a number of examples of hyperKähler cones, such as
quotient singularities (where the quantizations are spherical symplectic
reflection algebras) and nilpotent cones of simple Lie algebras (where the
quantizations are quotients of the enveloping algebra by a central character).
We also explain how short star-products on ${\mathcal{O}}(X)$ arise from
representation theory of its quantizations.

To summarize, short star-products appear to be a new structure in
representation theory, which is, in a sense, an algebraic incarnation of the
non-holomorphic ${\rm SU}(2)$-symmetry of a hyperKähler cone. While much of
representation theory is the study of quantizations of hyperKähler cones, this
symmetry has not been studied much by representation theorists, perhaps
because it is not algebraic (as in the context of representation theory,
hyperKähler cones are usually treated just as algebraic symplectic varieties).
However, the theory of short star-products suggests that the full hyperKähler
structure and the associated non-holomorphic ${\rm SU}(2)$ symmetry are also
quite relevant and worth taking into account in representation theory, as
pointed out in [1, footnote on p. 3]. The goal of this paper is to provide the
tools to do so, by connecting short star-products with more traditional and
well understood objects in representation theory (such as characters of
representations and Hochschild homology).

The organization of the paper is as follows. Section 2 contains preliminaries
on short star-products, filtered quantizations, quantization maps, the
evenness condition, and also some basics about symplectic singularities and
hyperKähler cones. In Section 3 we introduce nondegenerate twisted traces,
connect them to nondegenerate short star-products, and show that they depend
on a finite number of parameters, establishing our first main result. We also
discuss Hermitian and quaternionic structures on short star-product
quantizations, which are needed to study their positivity properties. Finally,
in Section 4 we consider a number of examples of hyperKähler cones and
construct and classify nondegenerate short star-products for them,
establishing our second main result. In particular, we establish rationality
of reduced characters of Verma modules (with arbitrary insertions) in category
$\mathcal{O}$ defined in [3, 21].

In the second part of this work, joint with Eric Rains, we plan to extend the
existence and classification of nondegenerate short star-products to a larger
class of hyperKähler cones (including Nakajima quiver varieties) and discuss
the relation with representation theory in more depth. We also plan to give
some explicit computations of short star-products for type A singularities,
connecting this work with the computational results of [1, 9], and study the
orthogonal polynomials that arise from them. Finally, we plan to study the
positivity properties of short star-products for type A singularities and in
general, fulfilling the program of [1].

Dedication (from Pavel Etingof). It is my great pleasure to dedicate this
paper to my teacher Dmitry Borisovich Fuchs on his 80th birthday. I have fond
memories of his topology lessons from over 30 years ago. Dmitry Borisovich
taught me to appreciate the beauty of interaction between different fields of
mathematics, which I enjoyed to a full extent while working on this paper.

##  2 Preliminaries

###  2.1 Star-product quantization of a graded Poisson algebra

Let $A$ be a commutative ${\mathbb{C}}$-algebra with a $\mathbb{Z}_{\geq 0}$
grading given by

$A=\bigoplus_{d\geq 0}A_{d}.$

Let $\\{~{},~{}\\}$ be a Poisson bracket of degree $-2$ on $A$; namely,

$\\{~{},~{}\\}\colon\ A_{n}\otimes A_{m}\to A_{n+m-2}.$

A basic example is the symmetric algebra $A=SV$ of a finite dimensional
symplectic vector space $V$ with the Poisson bracket defined by the symplectic
form.

Note that this setting includes the case of graded Poisson algebras with
Poisson bracket of degree $-1$ (such as the symmetric algebra of a Lie
algebra, the algebra of regular functions on the nilpotent cone, etc.), by
multiplying the degrees by $2$.444In fact, this theory can be developed for
Poisson brackets of any negative degree, but for most applications it is
sufficient to consider degree $-2$, so we restrict our attention to this case.

######  Definition 2.1.

A _star-product_ on $A$ quantizing $\\{\,,\,\\}$ is an associative
multiplication operation

$*\colon\ A\otimes A\to A$

such that for $a\in A_{n}$ and $b\in A_{m}$,

$a*b=\sum_{k=0}^{\lfloor\frac{n+m}{2}\rfloor}C_{k}(a,b),$

where $C_{k}\colon A_{n}\otimes A_{m}\to A_{n+m-2k}$ are bilinear maps such
that $C_{0}(a,b)=ab$ and

$\displaystyle C_{1}(a,b)-C_{1}(b,a)=\\{a,b\\}.$

######  Definition 2.2.

A star-product $*$ on $A$ is _short_ if for any $m,n\in{\mathbb{Z}}_{\geq 0}$
and any $a\in A_{n}$, $b\in A_{m}$ one has $C_{k}(a,b)=0$ for all
$k>\min(n,m)$. In other words, $a*b$ has no terms in $A_{d}$ for $d<|n-m|$.

From now on assume that $A_{0}={\mathbb{C}}$ and $\dim A_{i}<\infty$ for all
$i$. We define the following (in general, non-symmetric) inner product on $A$
induced by the star-product. Let $a\in A_{n}$ and $b\in A_{m}$. Then

$\langle a,b\rangle:=\text{CT}(a*b),$

where $\text{CT}\colon A\to A_{0}$ is the constant term map, taking the term
of degree zero. Note that if $*$ is short, then $\langle a,b\rangle=0$ if
$n\neq m$, i.e., the decomposition $A=\oplus_{d\geq 0}A_{d}$ is orthogonal.

######  Definition 2.3.

A short star-product $*$ on $A$ is said to be nondegenerate if the
corresponding inner product $\langle\,,\,\rangle$ is nondegenerate in each
degree $i$.

###  2.2 Construction of star-products from a quantum algebra

Let $A$ be a ${\mathbb{Z}}_{\geq 0}$-graded algebra. Fix a filtered
associative algebra

${{\mathbf{A}}}=\bigcup_{d\geq 0}F_{d}{{\mathbf{A}}}$

such that its associated graded algebra is identified with the graded algebra
$A$; namely,

$\operatorname{gr}{{\mathbf{A}}}=\bigoplus_{d\geq
0}F_{d}{{\mathbf{A}}}/F_{d-1}{{\mathbf{A}}}$

and $F_{d}{{\mathbf{A}}}/F_{d-1}{{\mathbf{A}}}\cong A_{d}$ for $d\geq 0$
(compatibly with multiplication) with $F_{-1}{{\mathbf{A}}}:=0$. Assume that
we have a filtration preserving ${\mathbb{Z}}/2$-action
$s\colon{{\mathbf{A}}}\to{{\mathbf{A}}}$ such that the corresponding
associated graded map $s\colon A\to A$ is given by $s=(-1)^{d}$, where
$d\colon A\to A$ is the degree operator. In this case, it is easy to see that
$[F_{n}{{\mathbf{A}}},F_{m}{{\mathbf{A}}}]\subset F_{n+m-2}{{\mathbf{A}}}$, so
we have the leading coefficient map $\\{\,,\,\\}\colon A_{n}\otimes A_{m}\to
A_{n+m-2}$. It is easy to check that $\\{\,,\,\\}$ is a Poisson bracket on
$A$, and ${{\mathbf{A}}}$ is a ${\mathbb{Z}}/2$-equivariant filtered
quantization of $\\{\,,\,\\}$.555Since all quantizations we will consider will
be ${\mathbb{Z}}/2$-equivariant, we will not mention it explicitly from now
on.

######  Definition 2.4.

A _quantization map_ is a ${\mathbb{Z}}/2$-equivariant isomorphism of vector
spaces

$\phi\colon\ A\to{{\mathbf{A}}}$

such that $\phi(A_{d})\subseteq F_{d}{{\mathbf{A}}}$ for $d\geq 0$ and
$\operatorname{gr}\phi\colon A\to A$ is the identity map, where

$\operatorname{gr}\phi=\bigoplus_{d\geq
0}\operatorname{gr}\phi_{d},\qquad\operatorname{gr}\phi_{d}\colon\ A_{d}\to
F_{d}{{\mathbf{A}}}/F_{d-1}{{\mathbf{A}}}=A_{d}.$

Note that $\phi$ need not (and usually cannot) be a homomorphism of algebras,
since the algebra $A$ is commutative but ${{\mathbf{A}}}$ is not in general.

Given a quantization map $\phi\colon A\to{{\mathbf{A}}}$, we can define the
star-product on $A$ by

$a*b:=\phi^{-1}(\phi(a)\phi(b)).$

Conversely, given a star-product on $A$, we can set ${{\mathbf{A}}}:=(A,*)$
with the filtration induced by the grading, and $\phi:={\rm Id}$.

######  Proposition 2.5.

This defines a pair of mutually inverse bijections between star-products on
$A$ and ${\mathbb{Z}}/2$-equivariant quantizations of $A$ equipped with a
quantization map.

###### Proof.

Straightforward. ∎

###  2.3 Even star-products

######  Definition 2.6.

A star-product $*$ on $A$ is said to be even if
$C_{k}(a,b)=(-1)^{k}C_{k}(b,a)$ for all $k\geq 0$.

In particular, for even star-products $C_{1}$ is skew-symmetric, so since
$C_{1}(a,b)-C_{1}(b,a)=\\{a,b\\}$, we have $C_{1}(a,b)=\frac{1}{2}\\{a,b\\}$.

Let $*$ be an even star-product on $A$ and ${{\mathbf{A}}}=(A,*)$ be the
corresponding quantum algebra. Then the map $\sigma={\rm i}^{d}$ defines an
antiautomorphism ${{\mathbf{A}}}\to{{\mathbf{A}}}$ such that
$\sigma^{2}=s=(-1)^{d}$. A ${\mathbb{Z}}/2$-invariant quantization
${\mathbf{A}}$ of $A$ equipped with a filtration-preserving antiautomorphism
$\sigma$ such that $\sigma^{2}=s$ will be called even. Thus an even star-
product gives rise to an even quantization. Conversely, an even quantization
${{\mathbf{A}}}$ of $A$ gives rise to an even star-product on $A$.

######  Proposition 2.7.

This defines a pair of mutually inverse bijections between even star-products
on $A$ and ${\mathbb{Z}}/2$-equivariant quantizations ${{\mathbf{A}}}$ of $A$
equipped with a filtered antiautomorphism
$\sigma\colon{{\mathbf{A}}}\to{{\mathbf{A}}}$ such that
$\operatorname{gr}\sigma={\rm i}^{d}$ and $\sigma^{2}=s$ with a
$\sigma$-invariant $($i.e., ${\mathbb{Z}}/4$-invariant$)$ quantization map.

###### Proof.

Straightforward. ∎

######  Example 2.8.

Let $A={\mathbb{C}}[x,y]$ with the Poisson bracket defined by $\\{y,x\\}=1$.
Let ${{\mathbf{A}}}$ be the Weyl algebra generated by $X$, $Y$ with the
defining relation $YX-XY=1$. This is a ${\mathbb{Z}}/2$-equivariant filtered
quantization of $A$ with the filtration defined by $\deg(X)=\deg(Y)=1$. Define
the map $\phi\colon A\to{{\mathbf{A}}}$ which sends $x^{i}y^{j}$ to the
average of all the orderings of the monomial $X^{i}Y^{j}$; in other words, it
is determined by the formula $\phi((px+qy)^{n})=(pX+qY)^{n}$ for any
$p,q\in{\mathbb{C}}$. For example, $\phi(1)=1$, $\phi(x)=X$, $\phi(y)=Y$,
$\phi(xy)=\frac{XY+YX}{2}=XY+\frac{1}{2}$. It is easy to show that this map
gives rise to the symmetric Moyal–Weyl star-product

$\displaystyle
a*b=\mu\left(\exp\left(\frac{\partial_{y}\otimes\partial_{x}-\partial_{x}\otimes\partial_{y}}{2}\right)(a\otimes
b)\right),$

where $\mu\colon A\otimes A\to A$ is the commutative multiplication. For
example,

$\displaystyle x*y=\phi^{-1}(XY)=xy-\frac{1}{2}.$

It is easy to check that this star-product is even, short, and nondegenerate.

The same statement holds in several variables, on a finite dimensional
symplectic vector space $V$ (i.e., for $A=SV$), where the symmetric Moyal–Weyl
star-product has the form

$\displaystyle a*b=\mu(\exp(\pi/2)(a\otimes b)),$

where $\pi\in\wedge^{2}V$ is the Poisson bivector.

More generally, given an element $\eta\in S^{2}V$, we may consider the short
star-product

$\displaystyle a*b=\mu(\exp((\eta+\pi)/2)(a\otimes b)),$

(the non-symmetric Moyal–Weyl star-product), which is not even if $\eta\neq
0$. Let $B\colon V\to V$ be the linear operator such that $\eta=(B\otimes
1)\pi$, i.e.,

$\displaystyle a*b=\mu(\exp(((B+1)\otimes 1)\pi/2)(a\otimes b));$ (2.1)

then $B\in\mathfrak{sp}(V)$. It is easy to show that $*$ is nondegenerate if
and only if the operator $B+1$ (or, equivalently, $B-1$) is invertible.

###  2.4 Conical symplectic singularities

The main class of examples of algebras $A$ we will be interested in is
$A={\mathcal{O}}(X)$, where $X$ is a (normal) conical symplectic singularity
in the sense of Beauville (see [17, 19] and references therein for their
definition and basic properties).

We will need the following lemma.

######  Lemma 2.9 ([19, Proposition 2.5]).

If $X$ is a conical symplectic singularity then every Poisson derivation of
$A:=\mathcal{O}(X)$ is inner.

###### Proof.

Here is another proof. Any such homogeneous derivation is defined by an (a
priori multivalued) holomorphic Hamiltonian $H$ on the big symplectic leaf
$X^{\circ}$ of $X$. Moreover, $H$ is, in fact, single-valued, since by [26]
the algebraic fundamental group of $X^{\circ}$ is finite (indeed, we have a
homomorphism $\phi\colon\pi_{1}(X^{\circ},x_{0})\to{\mathbb{C}}$ given by
$\phi(\gamma)=\gamma(H)-H$, where $\gamma(H)$ is the analytic continuation of
$H$ along $\gamma$, and $\phi=0$ since $\pi_{1}(X^{\circ},x_{0})_{\rm alg}$ is
finite). Then by normality of $X$ the function $H$ extends to the whole $X$,
to a homogeneous holomorphic function on $X$. This function is a holomorphic
section of a line bundle on ${\mathbb{P}}X$, which is algebraic by the GAGA
theorem, so $H\in\mathcal{O}(X)$, as claimed. ∎

It is shown in [19] that for conical symplectic singularities there is a very
nice parametrization of filtered quantizations of $A$, which is the same as
Namikawa’s parametrization of filtered Poisson deformations $X$ [25]. Namely,
they are parametrized by the finite dimensional space
$\mathfrak{P}=H^{2}\big{(}\widetilde{X}^{\rm reg},{\mathbb{C}}\big{)}$, where
$\widetilde{X}^{\rm reg}$ is the smooth part of the
${\mathbb{Q}}$-terminalization $\widetilde{X}$ of $X$, modulo the action of a
finite real reflection group $W$ called the Namikawa Weyl group. Denote the
quantization corresponding to $\lambda\in\mathfrak{P}$ by
${\mathbf{A}}_{\lambda}$. It is easy to show that ${\mathbf{A}}_{\lambda}$ is
${\mathbb{Z}}/2$-invariant.

Since $W$ is a reflection group, the space $\mathfrak{P}/W$ bijectively
parametrizing quantizations is an affine space, whose coordinates are
independent homogeneous generating invariants
$p_{1},\dots,p_{r}\in{\mathbb{C}}[\mathfrak{P}]^{W}$ of degrees
$d_{1},\dots,d_{r}$.

It is not hard to see that ${\mathbf{A}}_{\lambda}$ is even if and only if
$\lambda$ and $-\lambda$ are equivalent with respect to the Namikawa Weyl
group, i.e., if and only if there is an element $w\in W$ such that
$w\lambda=-\lambda$ (see [1]). This is equivalent to saying that $p_{i}=0$
whenever $d_{i}$ is odd, which defines a subspace in $\mathfrak{P}/W$.

######  Remark 2.10.

Lemma 2.9 implies that in the case of conical symplectic singularities the
antiautomorphism $\sigma\colon{\mathbf{A}}\to{\mathbf{A}}$ such that
$\sigma^{2}=s$ and $\operatorname{gr}\sigma={\rm i}^{d}$ is unique if exists.
Indeed, if $\sigma_{1}$, $\sigma_{2}$ are two such antiautomorphisms then
$g:=\sigma_{1}\circ\sigma_{2}^{-1}$ is an automorphism commuting with $s$ such
that $\operatorname{gr}g=1$. Hence $\log(g)$ is a linear combination of
derivations of ${\mathbf{A}}$ of degree $\leq-2$. If $g\neq 1$, then in the
leading order $\log(g)$ yields a nonzero homogeneous derivation $D\colon A\to
A$ of degree $\leq-2$. By Lemma 2.9, it is inner, so given by a Hamiltonian of
degree $\leq 0$. But such a Hamiltonian is necessarily constant, so $D=0$, a
contradiction.

The same argument proves that if $g\colon{\mathbf{A}}\to{\mathbf{A}}$ is a
filtration-preserving automorphism commuting with $s$ then $g$ is completely
determined by $\operatorname{gr}g$.

###  2.5 HyperKähler cones

We will be especially interested in conical symplectic singularities which are
hyperKähler cones. Such singularities play an important role in physics, since
many of them arise as Higgs and Coulomb branches in 3-dimensional $N=4$
superconformal field theories.

Namely, by a hyperKähler cone we mean a normal conical Poisson variety $X$
with Poisson bracket of negative degree (symplectic outside a set of
codimension $\geq 2$) and a hyperKähler structure on the smooth part that is
compatible to the Poisson structure (i.e., the complex structure on $X$ is
given by the operator $I$ and the Poisson structure by the symplectic form
$\omega:=\omega_{J}+{\rm i}\omega_{K}$). Like smooth hyperKähler manifolds,
hyperKähler cones are studied using their twistor space, which is the
universal family of complex structures on $X$ fibered over
${\mathbb{C}}{\mathbb{P}}^{1}$. The literature (especially mathematical) on
hyperKähler cones is rather scarce, but more details can be found in [4, 12,
31].666Note that the papers [4, 31] only deal with the smooth part of $X$.
However, it is expected that the geometry of $X$ is fully controlled by what
happens on the smooth part.

An important property of hyperKähler cones, which lies behind the phenomena
described in this paper, is that the complex structures corresponding to all
the points of ${\mathbb{C}}{\mathbb{P}}^{1}$ are equivalent, and in fact
permuted by a group ${\rm SU}(2)$ acting real analytically on $X$ (and by
rotations on ${\mathbb{C}}{\mathbb{P}}^{1}=S^{2}$). Thus the twistor space of
$X$ has the form $({\rm SU}(2)\times X)/{\rm U}(1)$, where the action of ${\rm
U}(1)$ on $X$ comes from the grading on $\mathcal{O}(X)$.

Important examples of hyperKähler cones include closures of nilpotent orbits
of a semisimple Lie algebra, Slodowy slices, quotients of a symplectic vector
space by a finite group of symplectic automorphisms, and more generally
hyperKähler reductions of a symplectic vector space by a compact Lie group of
symmetries (Higgs branches), which in particular include Nakajima quiver
varieties. Coulomb branches (when they are conical) are also expected to be
hyperKähler cones [24].

It is expected that under reasonable assumptions hyperKähler cones and in
particular Higgs and Coulomb branches should be conical symplectic
singularities, and this is known for Nakajima quiver varieties [2]. Also,
sufficient conditions for a Higgs branch to be a symplectic singularity (which
are satisfied in many examples) are given in [15]. So when we speak about
hyperKähler cones, we will assume that they are conical symplectic
singularities. However, it must be noted that certain nilpotent orbit closures
of classical groups fail to be symplectic singularities since they are not
normal, even though they are sometimes Higgs branches [18].

###  2.6 Finiteness of the number of fixed sets

######  Lemma 2.11.

Let $X$ be an affine scheme of finite type over ${\mathbb{C}}$ and $G$ be a
reductive group acting on $X$. Then up to the $G$-action there are finitely
many subschemes of $X$ of the form $X^{g}$, where $g\in G$ is a semisimple
element, .

###### Proof.

First consider the case when $G$ is diagonalizable (in which case every $g\in
G$ is semisimple). In this case the lemma is standard (see, e.g., [30,
Proposition on p. 44]). Namely, pick generators $f_{1},\dots,f_{n}$ of
${\mathbb{C}}[X]$ such that $f_{i}(gx)=\chi_{i}(g)f_{i}(x)$ for some
characters $\chi_{i}$ of $G$. Then
$f_{i}(gx)-f_{i}(x)=(\chi_{i}(g)-1)f_{i}(x)$, so $X^{g}$ is cut out by the
equations $f_{i}(x)=0$ for those $i$ for which $\chi_{i}(g)\neq 1$, which
implies that there are finitely many subschemes $X^{g}$.

The general case reduces to the diagonalizable case by using the following
fact (see [6] and [23, Section 1.14]): for each connected component $Z$ of $G$
there is a diagonalizable subgroup $D_{Z}\subset G$ such that every semisimple
element $g\in Z$ is $G^{0}$-conjugate to an element of $D_{Z}$.777We are
grateful to G. Lusztig for this explanation. ∎

##  3 Classification of short star-products

In this section we present the classification of nondegenerate short star-
products, the idea of which was explained to the first author by Maxim
Kontsevich.

###  3.1 Twisted traces

Consider the setting of Section 2.2. Let
$g\colon{{\mathbf{A}}}\to{{\mathbf{A}}}$ be a filtration-preserving invertible
linear map, and let $T\colon{{\mathbf{A}}}\to{\mathbb{C}}$ be a $g$-twisted
trace, i.e., $T({\mathbf{a}}{\mathbf{b}})=T({\mathbf{b}}g({\mathbf{a}}))$ for
all ${\mathbf{a}},{\mathbf{b}}\in{{\mathbf{A}}}$. Note that taking
${\mathbf{b}}=1$, we get $T(g({\mathbf{a}}))=T({\mathbf{a}})$, i.e., $T$ is
$g$-invariant.

Define the bilinear form $(\,,\,)_{T}$ on ${\mathbf{A}}$ by the formula
$({\mathbf{a}},{\mathbf{b}})_{T}:=T({\mathbf{a}}{\mathbf{b}})$.

######  Lemma 3.1.

Let $K$ be the left kernel of the bilinear form $(\,,\,)_{T}$. Then $K$ is
also the right kernel of $(\,,\,)_{T}$, and a $g$-invariant two-sided ideal in
${{\mathbf{A}}}$. In particular, if ${\mathbf{A}}$ is a simple algebra and
$T\neq 0$ then $K=0$.

###### Proof.

Let $K^{\prime}$ be the right kernel of $(\,,\,)_{T}$. We have
${\mathbf{b}}\in K^{\prime}$ if and only if $T({\textbf{ab}})=0$ for all
${\mathbf{a}}\in{{\mathbf{A}}}$. This is equivalent to saying that
$T({\mathbf{b}}g({\mathbf{a}}))=0$ for all ${\mathbf{a}}\in{{\mathbf{A}}}$,
which happens if and only if ${\mathbf{b}}\in K$. Thus, $K^{\prime}=K$. Also
${\mathbf{a}}\in K$ if and only if $T({\mathbf{b}}g({\mathbf{a}}))=0$ for all
${\mathbf{b}}\in{\mathbf{A}}$, which happens if and only if
$g({\mathbf{a}})\in K$, hence $K$ is $g$-invariant. Finally, if
${\mathbf{a}}\in K$ and ${\mathbf{b}}\in{{\mathbf{A}}}$ then
$({\textbf{ab}},{\textbf{c}})_{T}=({\textbf{a}},{\textbf{bc}})_{T}=0$ and
$({\textbf{c}},{\textbf{ba}})_{T}=({\textbf{cb}},{\textbf{a}})_{T}=0$, so
${\textbf{ab}},{\textbf{ba}}\in K$, i.e., $K$ is a two-sided ideal, as
desired. ∎

By Lemma 3.1, it makes sense to speak of the kernel of $(\,,\,)_{T}$ (without
specifying if it is left or right).

######  Lemma 3.2.

If the kernel of $(\,,\,)_{T}$ vanishes then $g$ is an algebra automorphism.

###### Proof.

We have

$\displaystyle
T({\mathbf{c}}g({\mathbf{a}}{\mathbf{b}}))=T({\textbf{abc}})=T({\textbf{bc}}g({\mathbf{a}}))=T({\mathbf{c}}g({\mathbf{a}})g({\mathbf{b}})),$

hence $g({\mathbf{ab}})=g({\mathbf{a}})g({\mathbf{b}})$. ∎

###  3.2 Nondegenerate short star-products and nondegenerate twisted traces

We will say that $T$ is nondegenerate if the restriction of the bilinear form
$({\mathbf{a}},{\mathbf{b}})_{T}:=T({\mathbf{a}}{\mathbf{b}})$ to
$F_{i}{{\mathbf{A}}}$ is nondegenerate for each $i$. Note that this is a
strictly stronger condition than vanishing of the kernel of $(\,,\,)_{T}$ (for
example, it implies that $T(1)\neq 0$).

Suppose that $T$ is nondegenerate and $s$-invariant (in this case $g$ commutes
with $s$). Then the bilinear form $(\,,\,)_{T}$ defines a canonical orthogonal
complement ${\mathbf{A}}^{T}_{i}$ to $F_{i-1}{{\mathbf{A}}}$ in
$F_{i}{{\mathbf{A}}}$. Note that the right and left orthogonal complement
coincide since $g$ is filtration-preserving. We have a natural isomorphism
$\theta_{i}^{T}\colon{\mathbf{A}}_{i}^{T}\to A_{i}$. Define the quantization
map $\phi^{T}\colon A\to{{\mathbf{A}}}$ by
$\phi^{T}|_{A_{i}}:=\big{(}\theta_{i}^{T}\big{)}^{-1}$. Clearly, $\phi^{T}$ is
$s$-invariant. Thus, $\phi^{T}$ defines a star-product $*$ on $A$. Clearly,
this star-product does not depend on the normalization of $T$, so we will
always normalize $T$ so that $T(1)=1$ (which is possible since for a
nondegenerate $T$, one has $T(1)\neq 0$).

######  Proposition 3.3 (M. Kontsevich).

The star-product $*$ is short and nondegenerate. Moreover, any nondegenerate
short star-product on $A$ is obtained in this way from a unique nondegenerate
$T$ such that $T(1)=1$. In other words, this correspondence is a bijection
between short nondegenerate star-products on $A$ and filtered quantizations
${{\mathbf{A}}}$ of $A$ equipped with a filtration-preserving automorphism $g$
commuting with $s$ and a nondegenerate $s$-invariant $g$-twisted trace $T$
such that $T(1)=1$.

###### Proof.

We have a nondegenerate (in general, nonsymmetric) inner product on $A$ given
by $\beta(a,b):=(\phi(a),\phi(b))_{T}$, and the spaces $A_{i}$ are orthogonal
under this inner product. Moreover, it is clear that
$\beta(a*b,c)=\beta(a,b*c)$. Suppose $a\in A_{m}$, $b\in A_{n}$. Let us show
that every homogeneous component of $a*b$ has degree $\geq|m-n|$. First assume
that $m>n$. It suffices to show that for any homogeneous $c\in A$ of degree
$d<m-n$, one has $\beta(a*b,c)=0$. But we have $\beta(a*b,c)=\beta(a,b*c)$,
and $b*c$ does not have any terms of degree bigger than $n+d<m$. Hence
$\beta(a*b,c)=\beta(a,b*c)=0$, as desired. A similar argument applies if
$m<n$: we have $\beta(c,a*b)=\beta(c*a,b)=0$. Thus, the star-product $*$ is
short. Moreover, we have

$\displaystyle\langle a,b\rangle={\rm CT}(a*b)=\beta(a*b,1)=\beta(a,b),$

which implies that $\langle\,,\,\rangle$ (and hence $*$) is nondegenerate.

Conversely, assume that $*$ is a nondegenerate short star-product on $A$. Then
the quantization ${{\mathbf{A}}}$ is identified with $(A,*)$ using the
quantization map $\phi$. Set $T({\mathbf{a}}):={\rm
CT}\big{(}\phi^{-1}({\mathbf{a}})\big{)}$. Then the form
$({\textbf{a}},{\textbf{b}})_{T}:=T({\textbf{ab}})$ is nondegenerate on each
$F_{i}{{\mathbf{A}}}$. Thus there exists a unique linear automorphism $g_{i}$
of $F_{i}{{\mathbf{A}}}$ such that
$T({\textbf{ab}})=T({\mathbf{b}}g_{i}({\mathbf{a}}))$. Moreover, it is clear
that $g_{i}$ preserves $F_{i-1}{{\mathbf{A}}}$, and
$g_{i}|_{F_{i-1}{{\mathbf{A}}}}=g_{i-1}$. Thus, the maps $g_{i}$ define a
filtration-preserving linear automorphism
$g\colon{{\mathbf{A}}}\to{{\mathbf{A}}}$.

Finally, it is easy to check that these assignments are mutually inverse, as
claimed. ∎

######  Remark 3.4.

If $T$ is any $s$-invariant linear functional on ${\mathbf{A}}$ such that the
form $({\mathbf{a}},{\mathbf{b}})_{T}=T({\mathbf{a}}{\mathbf{b}})$ is
nondegenerate on each $F_{i}{\mathbf{A}}$ (which clearly happens for “random”
$T$) then for each $i$ we can define a linear automorphism $g_{i}$ of
$F_{i}({\mathbf{A}})$ such that
$T({\textbf{ab}})=T({\mathbf{b}}g_{i}({\mathbf{a}}))$. However, in general
$g_{i}|_{F_{i-1}{\mathbf{A}}}\neq g_{i-1}$ and $g_{i}$ does not preserve
$F_{i-1}{\mathbf{A}}$ (so the automorphism $g$ of ${\mathbf{A}}$ restricting
to $g_{i}$ on each $F_{i}{\mathbf{A}}$ is not defined). Thus, the left and
right orthogonal complements of $F_{i-1}{\mathbf{A}}$ in $F_{i}{\mathbf{A}}$
do not coincide, and one cannot define a quantization map giving a short star-
product. The condition that $g$ is well defined is a very strong restriction
of $T$, and we will see that it usually leads to a finite dimensional space of
possible $T$.

Let ${\mathbf{A}}g$ be the ${\mathbf{A}}$-bimodule which is ${\mathbf{A}}$ as
a left module and ${\mathbf{A}}$ with action twisted by $g$ as a right module.

######  Corollary 3.5.

Let $S=S({\mathbf{A}})$ be the set of nondegenerate short star-products
corresponding to a filtered quantization ${{\mathbf{A}}}$ of $A$. Then we have
a map $\pi\colon S\to\operatorname{Aut}({{\mathbf{A}}})$ from $S$ to the group
of filtration-preserving $s$-invariant automorphisms of ${{\mathbf{A}}}$ whose
fiber $\pi^{-1}(g)$ is a subset of the space
$(HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}g)^{s})^{*}$ dual to the $s$-invariants
in the Hochschild homology $HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}g)$.

###### Proof.

This follows immediately from Proposition 3.3, since by definition, an
$s$-invariant $g$-twisted trace on ${{\mathbf{A}}}$ is an element of
$(HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}g)^{s})^{*}$. ∎

######  Corollary 3.6.

Suppose $A=\mathcal{O}(X)$, where $X$ is a conical symplectic singularity.
Then any nondegenerate short star-product corresponding to
$g\in\operatorname{Aut}({{\mathbf{A}}})$ is invariant with respect to the
connected component $Z_{g}^{0}$ of the centralizer $Z_{g}$ of $g$ in
$\operatorname{Aut}({{\mathbf{A}}})$. In particular, any nondegenerate even
short star-product is invariant under $\operatorname{Aut}({\mathbf{A}})^{0}$.

###### Proof.

It suffices to show that the Lie algebra $\operatorname{Lie}Z_{g}$ acts
trivially on $HH_{0}({\mathbf{A}},{\mathbf{A}}g)$. Let
$z\in\operatorname{Lie}Z_{g}$. By Lemma 2.9, there exists a unique
${\mathbf{z}}\in F_{2}{\mathbf{A}}$ up to adding a constant such that
$z({\mathbf{a}})=[{\mathbf{z}},{\mathbf{a}}]$ for all
${\mathbf{a}}\in{\mathbf{A}}$. Then $g({\mathbf{z}})={\mathbf{z}}+C$ for some
constant $C$, so $C=g({\mathbf{z}})-{\mathbf{z}}$. Thus if $C\neq 0$ then for
any $g$-twisted trace $T$ we have $T(1)=0$, so there are no nondegenerate
short star-products at all. On the other hand, if $C=0$ then we have
$[{\mathbf{z}},{\mathbf{a}}]={\mathbf{z}}{\mathbf{a}}-{\mathbf{a}}g({\mathbf{z}})$,
so ${\mathbf{z}}$ acts trivially on $HH_{0}({\mathbf{A}},{\mathbf{A}}g)$, as
claimed. The last statement follows from the fact that in the even case $g=s$,
and $s$ by definition is central in $\operatorname{Aut}({{\mathbf{A}}})$. ∎

###  3.3 The even case

######  Proposition 3.7.

A short nondegenerate star-product $*$ on $A$ is even if and only if the
corresponding automorphism $g$ of the associated even quantization
${\mathbf{A}}$ of $A$ equals $s$ and the corresponding nondegenerate trace $T$
on ${\mathbf{A}}$ is $\sigma$-invariant.

###### Proof.

Suppose $a,b\in A$, $\deg(a)=\deg(b)=d$. Then if $*$ is even then

$\displaystyle{\rm
CT}(a*b)=C_{d}(a,b)=(-1)^{d}C_{d}(b,a)=C_{d}\big{(}b,(-1)^{d}a\big{)}={\rm
CT}\big{(}b*(-1)^{d}a\big{)},$

hence $g=(-1)^{d}=s$. Also $T$ is $\sigma$-invariant since so is $\phi$.

Conversely, suppose $g=s$ and $T$ is $\sigma$-invariant. Let $a\in A_{m}$,
$b\in A_{n}$, $c\in A_{m+n-2k}$. Let $\phi(a)={\mathbf{a}}$,
$\phi(b)={\mathbf{b}}$, $\phi(c)={\mathbf{c}}$. We have

$\displaystyle
T(\phi(a*b){\mathbf{c}})=T({\mathbf{a}}{\mathbf{b}}{\mathbf{c}})=T(\sigma({\mathbf{a}}{\mathbf{b}}{\mathbf{c}}))=T(\sigma({\mathbf{c}})\sigma({\mathbf{b}})\sigma({\mathbf{a}}))$
$\displaystyle\hphantom{T(\phi(a*b){\mathbf{c}})}{}=T\big{(}\sigma({\mathbf{b}})\sigma({\mathbf{a}})\sigma^{-1}({\mathbf{c}})\big{)}=(-1)^{k}T({\mathbf{b}}{\mathbf{a}}{\mathbf{c}})=T\big{(}\phi\big{(}(-1)^{k}b*a\big{)}{\mathbf{c}}\big{)}.$

Thus $C_{k}(a,b)=(-1)^{k}C_{k}(b,a)$, i.e., $*$ is even. ∎

Let ${{\mathbf{A}}}_{\pm}$ be the $\pm 1$-eigenspaces of $s$ on
${{\mathbf{A}}}$.

######  Corollary 3.8.

A star-product $*$ is even if and only if $T$ is $\sigma$-stable and defines
an element of $(HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}s)^{s})^{*}$, where

$\displaystyle
HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}s)^{s}=({{\mathbf{A}}}s/[{{\mathbf{A}}},{{\mathbf{A}}}s])^{s}={{\mathbf{A}}}_{+}/([{{\mathbf{A}}}_{+},{{\mathbf{A}}}_{+}]+\\{{{{\mathbf{A}}}_{-},{{\mathbf{A}}}_{-}\\}}),$

and $\\{{{\mathbf{a}},{\mathbf{b}}\\}}:={\mathbf{ab}}+{\mathbf{ba}}$.

###### Proof.

This follows immediately from Corollary 3.5 and Proposition 3.7. ∎

###  3.4 Examples

######  Example 3.9.

Let $V$ be a finite dimensional symplectic vector space. Let us classify
nondegenerate short star-products for the polynomial algebra $A=SV$ with the
standard grading and Poisson structure. The quantization ${{\mathbf{A}}}$ in
this case is the Weyl algebra ${\mathbf{W}}(V)$. Its group of filtered
automorphisms commuting with $s$ is $G={\rm Sp}(V)$.

Let $g\in G$, and let us compute the space
${\mathcal{T}}:=HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}g)^{*}$. First, we claim
that if $g$ has eigenvalue $1$ then this space vanishes. Indeed, let $u\in V$
be a nonzero vector such that $gu=u$. Then for $T\in{\mathcal{T}}$ we have
$T([u,{\mathbf{b}}])=0$ for all ${\mathbf{b}}\in{\mathbf{A}}$. It is easy to
see that any element ${\mathbf{a}}\in{{\mathbf{A}}}$ can be written as
$[u,{\mathbf{b}}]$. Thus, $T=0$, as desired.

Now consider the case when $g$ has no eigenvalue $1$. In this case,
$T({\mathbf{ab}}-{\mathbf{b}}g({\mathbf{a}}))=0$. Let $E\subset{{\mathbf{A}}}$
be the span of elements ${\mathbf{ab}}-{\mathbf{b}}g({\mathbf{a}})$ for any
${\mathbf{a}},{\mathbf{b}}\in{\mathbf{A}}$. Taking $a,b\in A$ homogeneous and
${\mathbf{a}}$, ${\mathbf{b}}$ their lifts to ${\mathbf{A}}$, we see that
$\operatorname{gr}E$ contains elements $(a-g(a))b$, where $a,b\in A$ are
homogeneous. Taking $\deg(a)=1$, we find that $\operatorname{gr}E$ contains
the augmentation ideal of $SV$, which shows that $\dim{\mathcal{T}}\leq 1$.

Let us show that in fact $\dim{\mathcal{T}}=1$. For this purpose it suffices
to produce a nonzero $g$-twisted trace. To this end, let $B:=(1+g)(1-g)^{-1}$
be the (symplectic) Cayley transform of $g$; then $B\in{\mathfrak{sp}}(V)$ and
$g=(B+1)^{-1}(B-1)$, so $B+1$ and $B-1$ are invertible. Hence we have the
Moyal–Weyl product given by (2.1), which is short and nondegenerate. As we
have shown, such a star-product defines a nondegenerate $g$-twisted trace $T$.
Thus we have $\mathcal{T}=\mathcal{T}^{s}={\mathbb{C}}T$, a 1-dimensional
space, i.e., we have a unique star-product for each $g$.

In summary, we see that short nondegenerate star-products on $SV$ are exactly
the Moyal–Weyl products parametrized by elements of $g\in{\rm Sp}(V)$ without
eigenvalue $1$, or, equivalently, elements of $B\in{\mathfrak{sp}}(V)$ without
eigenvalues $1$ and $-1$. The two parametrizations are related by the Cayley
transform. The even case corresponds to $g=-1$, i.e., $B=0$, i.e., the
symmetric Moyal–Weyl product is the unique even short nondegenerate star-
product.

Here is an explicit construction of the corresponding twisted trace. Assume
first that $g$ has no eigenvalues of absolute value $1$. Let $L\subset V$ be
the direct sum of generalized eigenspaces of $g$ with eigenvalues $\lambda$
such that $|\lambda|<1$. Then the Weyl algebra ${\mathbf{A}}$ is naturally
identified with the algebra $D(L^{*})$ of differential operators on $L^{*}$,
which acts naturally on $M:=SL={\mathbb{C}}[L^{*}]$. Then the trace $T$ can be
defined by the formula

$\displaystyle
T({\mathbf{a}})=\frac{\operatorname{Tr}_{M}({\mathbf{a}}g)}{\operatorname{Tr}_{M}(g)}={\det}_{L}(1-g)\operatorname{Tr}_{M}({\mathbf{a}}g).$

Note that the trace is convergent since the eigenvalues of $g$ on $L$ have
absolute value $<1$. Moreover, it is easy to check that the right hand side is
a rational function of $g$ which is regular on the locus where $g$ has no
eigenvalue $1$. Thus the above formula understood as a rational function of
$g$ gives the desired twisted trace in the general case.

######  Example 3.10.

Let ${{\mathbf{A}}}=D_{\lambda}:={\rm U}(\mathfrak{sl}_{2})/I_{\lambda}$,
where $I_{\lambda}$ is the ideal generated by
$C-\frac{\lambda(\lambda+2)}{2}$, where $C=ef+fe+h^{2}/2$ is the Casimir. Then
$A$ is the algebra of regular functions on the quadratic cone
$X\subset{\mathbb{C}}^{3}$, $A={\mathbb{C}}[x,y,z]/\big{(}xy-z^{2}\big{)}$,
with the variables having degree $2$. Let us classify short nondegenerate
star-products giving this quantization.

Let us start with the even case. We have $s=1$ and
${\mathbf{A}}={\mathbf{A}}_{+}$, so by Corollary 3.8, the star-product gives
rise to a trace in $HH_{0}({\mathbf{A}},{\mathbf{A}})^{*}$, which is well
known to be 1-dimensional. This trace is therefore $\sigma$-stable and also
nondegenerate for Weil generic $\lambda$ (i.e., outside of a countable set),
since it is nondegenerate for $\lambda=-1/2$, when $D_{\lambda}$ is isomorphic
to the even part of the Weyl algebra, ${\mathbf{A}}_{1}^{{\mathbb{Z}}/2}$, so
we can use the symmetric Moyal–Weyl product. This nondegenerate trace gives
rise to the unique ${\rm PGL}(2)$-invariant short star-product, which is thus
even and nondegenerate.

However, for special values of $\lambda$ (namely, $\lambda\in{\mathbb{Z}}$,
$\lambda\neq-1$) this trace is degenerate. Indeed, we may assume
$\lambda\in{\mathbb{Z}}_{\geq 0}$, then the trace is given by the character of
the finite dimensional irreducible representation $L_{\lambda}$ of
${\mathfrak{sl}}_{2}$ with highest weight $\lambda$, and the annihilator of
this representation is the kernel of $(\,,\,)_{T}$. The short even star-
product still exists at these points (by continuity), namely it comes from the
unique ${\rm PGL}(2)$-invariant quantization map, but it is degenerate.

In fact, it is not hard to see that outside of these special points, the star-
product is nondegenerate. Indeed, it is clear that the bilinear form $C_{2m}$
on $A_{2m}$ is the standard invariant pairing $V_{2m}\otimes
V_{2m}\to{\mathbb{C}}$ times a polynomial $P_{m}(\lambda)$ of degree $2m$.
Moreover, as explained above, this polynomial vanishes at $0,\dots,m-1$ and
$-2,\dots,-m-1$, so it is a nonzero multiple of
$\prod\limits_{j=0}^{m-1}(\lambda-j)(\lambda+j+2)$, i.e., it does not vanish
at any other points, as claimed.

Now let $g\neq 1$. We have $\operatorname{Aut}({\mathbf{A}})={\rm
PGL}_{2}({\mathbb{C}})$, so $g\in{\rm PGL}_{2}({\mathbb{C}})$. First consider
the case when $g$ is semisimple (so we may assume that $g$ is in the standard
torus). Assume that $|\alpha(g)|>1$ for the positive root $\alpha$. Let
$M_{\lambda}$ be the Verma module over $\mathfrak{sl}_{2}$ with highest weight
$\lambda$, which is also a $D_{\lambda}$-module. Consider the linear
functional $T_{\lambda}^{w}$ on $D_{\lambda}$ given by

$\displaystyle
T_{\lambda}^{w}({\mathbf{a}}):=\frac{\operatorname{Tr}|_{M_{\lambda}}({\mathbf{a}}g)}{\operatorname{Tr}|_{M_{\lambda}}(g)}=(1-w)\sum_{n\geq
0}(v_{n}^{*},{\mathbf{a}}v_{n})w^{n},$

where $w=\alpha(g)^{-1}$, $v_{n}=f^{n}v$ is a homogeneous basis of
$M_{\lambda}$ (where $v$ is a highest weight vector) and $v_{n}^{*}$ is the
dual basis of the restricted dual space $M_{\lambda}^{\vee}$. It is easy to
show that this series is convergent to a rational function of $w$ regular for
$w\neq 1$, and thus defines a $g$-twisted trace on $D_{\lambda}$. We also have
the $g$-twisted trace $T_{-\lambda-2}^{w}$, since by definition
$D_{\lambda}=D_{-\lambda-2}$. Moreover, we have
$\mathcal{O}(X^{g})={\mathbb{C}}[z]/\big{(}z^{2}\big{)}$, so by Proposition
3.11 below, the space of $g$-twisted traces
$HH_{0}({\mathbf{A}},{\mathbf{A}}g)^{*}$ is at most 2-dimensional. On the
other hand, for $\lambda\neq-1$, the traces $T_{\lambda}^{w}$ and
$T_{-\lambda-2}^{w}$ are linearly independent, since $T_{\lambda}^{w}(1)=1$
and $T_{\lambda}^{w}(h)=\lambda-\frac{2w}{1-w}$, thus they form a basis in the
space of $g$-twisted traces. Moreover, at $\lambda=-1$ we have the trace
$\big{(}T_{-1}^{w}\big{)}^{\prime}:=\lim\limits_{\varepsilon\to
0}\frac{T_{-1+\varepsilon}^{w}-T_{-1-\varepsilon}^{w}}{\varepsilon}$, and it
is easy to see that $T_{-1}^{w}$, $\big{(}T_{-1}^{w}\big{)}^{\prime}$ form a
basis in the space of traces when $\lambda=-1$.

Moreover, for $\lambda\neq-1$ the trace $T_{\lambda}$ corresponding $g=1$ can
be obtained by the formula

$\displaystyle T_{\lambda}({\mathbf{a}})=\frac{1}{\lambda+1}\lim_{w\to
1}\frac{T_{\lambda}^{w}({\mathbf{a}})-w^{\lambda+1}T_{-\lambda-2}^{w}({\mathbf{a}})}{1-w}.$

Indeed, if $\lambda$ is a nonnegative integer, this equals
$\frac{1}{\lambda+1}\operatorname{Tr}|_{L_{\lambda}}({\mathbf{a}})$. The case
$\lambda=-1$ is then obtained as a limit, using that this is in fact a
polynomial in $\lambda$. For example, for ${\mathbf{a}}=h^{2}$ we get

$\displaystyle\operatorname{Tr}|_{L_{\lambda}}\big{(}h^{2}\big{)}=2\sum_{0\leq
j\leq\lambda/2}(\lambda-2j)^{2}=\frac{\lambda(\lambda+1)(\lambda+2)}{3}.$

Thus

$\displaystyle T_{\lambda}\big{(}h^{2}\big{)}=\frac{\lambda(\lambda+2)}{3}.$

It is easy to show that Weil generically888Here “Weil generically” means
“outside of a countable union of algebraic hypersurfaces in the parameter
space”. these traces are nondegenerate. Thus we obtain a 3-parameter family of
nondegenerate $h$-invariant short star-products on $A$ inequivalent under
${\rm PGL}_{2}({\mathbb{C}})$, and any nondegenerate short star-product with
semisimple $g$ is equivalent to one of them. Namely, the parameters are
$\lambda$, $w$, and $\alpha:=T(h)$ (where $T$ is normalized, i.e., $T(1)=1$).

Finally, it remains to consider the case when $g$ is not semisimple (i.e., a
Jordan block). In this case, by Proposition 3.14 below, the space of traces is
at most 1-dimensional. If $\lambda$ is a nonnegative integer, the
corresponding trace can be given by the formula

$\displaystyle\widetilde{T}_{\lambda}({\mathbf{a}})=\frac{1}{\lambda+1}\operatorname{Tr}|_{L_{\lambda}}({\mathbf{a}}g)$

(where, abusing notation, we denote by $g$ the unipotent lift of $g$ to ${\rm
SL}_{2}({\mathbb{C}})$). It is not hard to check that this is in fact a
polynomial of $\lambda$, so it can be naturally extended to any value of
$\lambda$.

###  3.5 Finite dimensionality of the space of nondegenerate short star-
products

####  3.5.1 The case of a semisimple automorphism

######  Proposition 3.11.

Let $A$ be a finitely generated Poisson algebra such that the Poisson scheme
$X=\operatorname{Spec}A$ has finitely many symplectic leaves. Let
${\mathbf{A}}$ be a quantization of $A$, and
$g\colon{{\mathbf{A}}}\to{{\mathbf{A}}}$ be a semisimple automorphism which
commutes with $s$. Then nondegenerate short star-products on $A$, if they
exist, are parametrized by points of the finite dimensional vector space
$(HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}g)^{s})^{*}$ which don’t belong to a
countable collection of hypersurfaces, modulo scaling.

Moreover, $\dim HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}g)^{s}\leq\dim
HP_{0}(X^{g})^{s}<\infty$, where $X^{g}$ is the fixed point scheme of $g$ on
$X$, and
$HP_{0}(X^{g}):=\mathcal{O}(X^{g})/\\{{\mathcal{O}(X^{g}),\mathcal{O}(X^{g})\\}}$
is the zeroth Poisson homology of $X^{g}$.999Here, abusing notation, we denote
the associated graded isomorphism $\operatorname{gr}g\colon A\to A$ simply by
$g$.

###### Proof.

Let $E$ be the span of elements ${\mathbf{ab}}-{\mathbf{b}}g({\mathbf{a}})$
for ${\mathbf{a}},{\mathbf{b}}\in{{\mathbf{A}}}$.

We claim that for any $a,b\in A$, $\operatorname{gr}(E)$ contains the element
$(a-g(a))b$. Indeed, we may assume that $a$, $b$ are homogeneous of degrees
$i$, $j$, and let ${{\mathbf{a}}}$, ${\mathbf{b}}$ be their lifts to
${{\mathbf{A}}}$. Then the leading term (of degree $i+j$) of
${{\mathbf{a}}}{\mathbf{b}}-{\mathbf{b}}g({{\mathbf{a}}})$ is $(a-g(a))b$, as
desired.

Also, we claim that if $a\in A$ and $g(a)=a$ then for any $b\in A$,
$\\{{a,b\\}}$ belongs to $\operatorname{gr}(E)$. Indeed, we can again assume
that $a$, $b$ are homogeneous of degrees $i$, $j$. Since $g=1$ on its
generalized 1-eigenspace, we can choose a lift ${\mathbf{a}}$ of $a$ to
${\mathbf{A}}$ which is invariant under $g$. Then choosing any lift
${\mathbf{b}}$ of $b$ to ${\mathbf{A}}$, we see that the leading term (of
degree $i+j-2$) of
${{\mathbf{a}}}{\mathbf{b}}-{\mathbf{b}}g({{\mathbf{a}}})=[{{\mathbf{a}}},{\mathbf{b}}]$
is $\\{{a,b\\}}$, as desired.

Now let $I\subset A$ be the span of elements $(a-g(a))b$. This is the ideal of
the fixed point subscheme $X^{g}$. Note that $I\cap A^{g}$ is a Poisson ideal
in $A^{g}$. Hence, $A/I=A^{g}/(I\cap A^{g})$ is a Poisson algebra, so $X^{g}$
is a Poisson scheme. Moreover, since $g=1$ on its generalized 1-eigenspace,
the $g$-invariants in the tangent space to the symplectic leaf $L$ at any
point $P\in X^{g}\cap L$ form a nondegenerate subspace, hence $X^{g}\cap L$ is
symplectic. Thus, $X^{g}$ also has finitely many symplectic leaves, which are
connected components of intersections of $X^{g}\cap L$ for various $L$.

Now, the space $A/\operatorname{gr}(E)$ receives a surjective $s$-invariant
map from the space
$HP_{0}(X^{g})=\mathcal{O}(X^{g})/\\{{\mathcal{O}(X^{g}),\mathcal{O}(X^{g})\\}}$.
This space is finite dimensional by [11, Theorem 1.4]. Hence the space
$A/\operatorname{gr}(E)$ is finite dimensional. Therefore the space
${{\mathbf{A}}}/E=HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}g)$ is finite
dimensional, and we have the inequalities

$\displaystyle\dim HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}g)\leq\dim
HP_{0}(X^{g}),\qquad\dim HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}g)^{s}\leq\dim
HP_{0}(X^{g})^{s}.$

Now the proposition follows, since by Corollary 3.5, short nondegenerate star-
products corresponding to $g$ are parametrized by the subset of
$(HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}g)^{s})^{*}$ determined by the
nondegeneracy condition, which is the vanishing condition for countably many
determinant polynomials. ∎

######  Remark 3.12.

  1. 1.

It is clear from the proof of Proposition 3.11 that it holds more generally,
namely when $g$ acts trivially on its generalized 1-eigenspace (this is the
only consequence of semisimplicity of $g$ used in the proof).

  2. 2.

In particular, Proposition 3.11 applies to the setting of [19], namely
$A=\mathcal{O}(X)$, where $X$ is a conical symplectic singularity in the sense
of Beauville. Indeed, in this case it is known that $X$ has finitely many
symplectic leaves [17, Theorem 2.3]. This property also holds for Higgs
branches considered in [1] (by [20, Section 2.3]), and is expected for Coulomb
branches which are cones, so Proposition 3.11 applies to all such cases.

######  Corollary 3.13.

Let $A$ be a finitely generated algebra, and the Poisson scheme
$X=\operatorname{Spec}A$ have finitely many symplectic leaves. Then
nondegenerate even short star-products on $A$ defining a given even
quantization ${{\mathbf{A}}}$, if they exist, are parametrized by points of
the finite dimensional vector space
$(HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}s)^{\sigma})^{*}$ which don’t belong to
a countable collection of hypersurfaces, modulo scaling.

Moreover, $\dim HH_{0}({{\mathbf{A}}},{{\mathbf{A}}}s)^{\sigma}\leq\dim
HP_{0}(X^{s})^{\sigma}<\infty$.

###### Proof.

This immediately follows from Proposition 3.11, specializing to the case $g=s$
and restricting to $\sigma$-invariant traces using Corollary 3.8. ∎

Corollary 3.13 shows that if $\operatorname{Spec}A$ has finitely many
symplectic leaves then nondegenerate short even star-products depend on
finitely many parameters, as conjectured in [1].

####  3.5.2 The general case

Now let us generalize Proposition 3.11 to the nonsemisimple case. For this
purpose assume that every Poisson derivation of $A$ of nonpositive degree is
inner. Then the Lie algebra $\mathfrak{g}$ of the group
$G:=\operatorname{Aut}_{0}(A)$ of filtration-preserving Poisson automorphisms
of $A$ commuting with $s$ has the form $\mathfrak{g}=A_{2}$, where the
operation is the Poisson bracket.

Let us also assume that the group $G$ is reductive. Then it is easy to see
that the action of $\mathfrak{g}$ on $A$ uniquely lifts to any quantization
${\mathbf{A}}$, so we have ${\rm Der}({\mathbf{A}})^{s}=\mathfrak{g}$. Thus we
have a Lie algebra inclusion $\mathfrak{g}\subset
F_{2}{\mathbf{A}}\subset{\mathbf{A}}$.

Let $g\in G$. Write $g=g_{0}\exp(e)=\exp(e)g_{0}$ where $g_{0}\in G$ is
semisimple and $e\in\mathfrak{g}$ is nilpotent.

######  Proposition 3.14.

Under the above assumptions, the conclusion of Proposition 3.11 holds for $g$.
Moreover,

$\displaystyle\dim HH_{0}({\mathbf{A}},{\mathbf{A}}g)^{s}\leq\dim
HH_{0}({\mathbf{A}},{\mathbf{A}}g_{0})^{s}\leq\dim
HP_{0}(X^{g_{0}})^{s}<\infty.$

###### Proof.

Let $Z$ be the centralizer of $g_{0}$ in $G$, a reductive group containing
$\exp(e)$. By the Jacobson–Morozov lemma, the Lie algebra $\mathfrak{z}$ of
$Z$ contains an $\mathfrak{sl}_{2}$-triple $e$, $h$, $f$. We have a
decomposition ${\mathbf{A}}=\oplus_{j\in{\mathbb{Z}}}{\mathbf{A}}_{j}$, where
${\mathbf{A}}_{j}$ is the $j$-eigenspace of $h$. Since $g(e)=e$, we have
$e{\mathbf{b}}-{\mathbf{b}}g(e)=[e,{\mathbf{b}}]$, so for any $T\in
HH_{0}({\mathbf{A}},{\mathbf{A}}g)^{*}$ and ${\mathbf{b}}\in{\mathbf{A}}$ we
have $T([e,{\mathbf{b}}])=0$. By representation theory of $\mathfrak{sl}_{2}$,
this implies that $T({\mathbf{a}})=0$ for any
${\mathbf{a}}\in{\mathbf{A}}_{j}$ with $j>0$.

Now consider the 1-parameter subgroup $t^{h}$ of $G$
($t\in{\mathbb{C}}^{\times}$), and define
$T_{t}({\mathbf{a}}):=T\big{(}t^{-h}{\mathbf{a}}t^{h}\big{)}$. If
${\mathbf{a}}\in{\mathbf{A}}_{j}$ then
$T_{t}({\mathbf{a}})=t^{-j}T({\mathbf{a}})$. Thus, since $T$ vanishes on
${\mathbf{A}}_{j}$ for $j>0$, there exists a limit $T_{0}=\lim\limits_{t\to
0}T_{t}$, which coincides with $T$ on ${\mathbf{A}}_{0}$ and vanishes on
${\mathbf{A}}_{j}$ for $j\neq 0$.

We have $T({\mathbf{a}})=T_{t}({\mathbf{a}}_{t})$, where
${\mathbf{a}}_{t}:=t^{h}{\mathbf{a}}t^{-h}$. Recall that

$\displaystyle{\mathbf{a}}{\mathbf{b}}-{\mathbf{b}}g({\mathbf{a}})={\mathbf{a}}{\mathbf{b}}-{\mathbf{b}}g_{0}({\mathbf{a}})-{\mathbf{b}}\sum_{i\geq
1}\frac{g_{0}e^{i}}{i!}({\mathbf{a}})\in\operatorname{Ker}(T).$

Hence

$\displaystyle
t^{h}({\mathbf{a}}{\mathbf{b}}-{\mathbf{b}}g({\mathbf{a}}))t^{-h}={\mathbf{a}}_{t}{\mathbf{b}}_{t}-{\mathbf{b}}_{t}g_{0}({\mathbf{a}}_{t})-{\mathbf{b}}_{t}\sum_{i\geq
1}\frac{t^{2i}e^{i}}{i!}({\mathbf{a}}_{t})\in\operatorname{Ker}(T_{t}).$

Thus, for any ${\mathbf{a}},{\mathbf{b}}\in{\mathbf{A}}$ the element
${\mathbf{a}}{\mathbf{b}}-{\mathbf{b}}g_{0}({\mathbf{a}})-{\mathbf{b}}\sum\limits_{i\geq
1}\frac{t^{2i}g_{0}e^{i}}{i!}({\mathbf{a}})$ belongs to
$\operatorname{Ker}(T_{t})$. Sending $t$ to zero, we see that
${\mathbf{a}}{\mathbf{b}}-{\mathbf{b}}g_{0}({\mathbf{a}})\in\operatorname{Ker}(T_{0})$.
Thus, $T_{0}\in(HH_{0}({\mathbf{A}},{\mathbf{A}}g_{0})^{s})^{*}$, which is
finite dimensional (with dimension dominated by $\dim HP_{0}(X^{g_{0}})^{s}$)
by Proposition 3.11.

It remains to show that the assignment $T\mapsto T_{0}$ is injective; this
implies both statements of the theorem. To this end, assume that $T_{0}=0$ but
$T\neq 0$, and let $m>0$ be the order of vanishing of $T_{t}$ at $t=0$. Let
$T_{0}^{\prime}:=\lim\limits_{t\to 0}T_{t}/t^{m}$. Then $T_{0}^{\prime}$ is a
nonzero element of $HH_{0}({\mathbf{A}},{\mathbf{A}}g_{0})^{*}$ which vanishes
on ${\mathbf{A}}_{0}$. But if ${\mathbf{a}}\in{\mathbf{A}}_{j}$, $j\neq 0$,
then ${\mathbf{a}}=\frac{1}{j}[h,{\mathbf{a}}]$, so
$T_{0}^{\prime}({\mathbf{a}})=0$. This is a contradiction, which completes the
proof of the proposition. ∎

######  Theorem 3.15.

Let $A=\mathcal{O}(X)$, where $X$ is a conical symplectic singularity with
Poisson bracket of degree $-2$ such that the group $G$ of dilation-invariant
Poisson automorphisms of $X$ is reductive. Then for any quantization
${\mathbf{A}}$ of $A$ and any $g\in G$ the space
$HH_{0}({\mathbf{A}},{\mathbf{A}}g)^{s}$ is finite dimensional, with dimension
dominated by $\dim HP_{0}(X^{g_{0}})^{s}$, which is finite. Therefore,
nondegenerate short star-products on $A$ depend on finitely many parameters.

###### Proof.

The theorem follows immediately from the above results and Lemmas 2.9, 2.11. ∎

The group $G$ is not always reductive, see the example in the beginning of
[27]. However, this is true in most applications, in particular if $X$ is a
hyperKähler cone (the case considered in [1] and [9]), i.e., the holomorphic
symplectic structure on the big symplectic leaf on $X$ comes from a
hyperKähler structure (as complex structure $I$). In this case, by [16,
Theorem 3.3] (see also the discussion in [12, p. 22]), the hyperKähler metric
is determined by the holomorphic symplectic structure and the complex
conjugation map through the twistor construction (possibly up to finitely many
choices). This means that the real part $G^{0}_{{\mathbb{R}}}$ of the identity
component $G^{0}$ of the group $G$ preserves the hyperKähler metric. This
implies that $G^{0}_{{\mathbb{R}}}$ preserves the unitary structure on the
space $A_{d}$ for each $d$ given by

$\displaystyle||f||^{2}:=\int_{X}|f(z)|^{2}{\rm e}^{-D(z)}{\rm d}V,$

where ${\rm d}V$ is the volume element and $D(z)$ the distance to the origin
determined by the hyperKähler metric. This implies that $G^{0}_{{\mathbb{R}}}$
is compact, hence $G^{0}$ and $G$ are reductive, as desired.

Thus we obtain the following generalization of the conjecture from [1] to the
not necessarily even case, which is our first main result.

######  Theorem 3.16.

Theorem 3.15 holds for hyperKähler cones.

###  3.6 Conjugations

As before, let $A$ be a graded Poisson algebra with Poisson bracket of degree
$-2$.

######  Definition 3.17.

  1. (i)

A conjugation on $A$ is a ${\mathbb{C}}$-antilinear degree-preserving Poisson
automorphism $\rho\colon A\to A$.

  2. (ii)

If ${\mathbf{A}}$ is a quantization of $A$ then a conjugation on
${\mathbf{A}}$ is an antilinear filtration-preserving automorphism $\rho$ of
${\mathbf{A}}$ which commutes with $s$.

It is clear that any conjugation on ${\mathbf{A}}$ gives rise to a conjugation
on $A$ (by taking the associated graded). Also note that if $\rho$ is a
conjugation then $\rho^{2}$ is a ${\mathbb{C}}$-linear automorphism.

Let us say that the conjugations $\rho$ and $\rho^{\prime}$ are equivalent if
$\rho^{\prime}=h\rho h^{-1}$ for some automorphism $h$. Clearly, it makes
sense to distinguish conjugations only up to equivalence, since equivalent
conjugations are “the same for all practical purposes”.

######  Definition 3.18.

If $A$ is equipped with a conjugation $\rho$ then we call a star-product $*$
on $A$ conjugation-invariant if $\rho(a*b)=\rho(a)*\rho(b)$.

It is easy to see that conjugation-invariant star-products on $A$ correspond
to quantizations ${\mathbf{A}}$ of $A$ with a conjugation (whose associated
graded is the conjugation of $A$) equipped with a conjugation-invariant
quantization map $\phi$.

######  Definition 3.19.

We say a conjugation-invariant short star-product $*$ on $A$ is Hermitian if
$CT(a*b)=CT\big{(}b*\rho^{2}(a)\big{)}$ for $a,b\in A$.

This terminology is motivated by the following lemma.

######  Lemma 3.20.

Let $*$ be a conjugation-invariant short star-product on $A$. Then $*$ is
Hermitian if and only if the sesquilinear form $a,b\mapsto\langle
a,\rho(b)\rangle=CT(a*\rho(b))$ is Hermitian.

###### Proof.

We have

$\displaystyle\overline{CT\big{(}\rho^{-1}(b)*\rho(a)\big{)}}=CT\big{(}b*\rho^{2}(a)\big{)}.$

The Hermitian condition is that this equals $CT(a*b)$, as desired. ∎

######  Definition 3.21.

We will say that a Hermitian short star-product $*$ on $A$ is unitary if the
Hermitian form $\langle a,\rho(b)\rangle$ is positive definite (i.e., it is
positive definite on $A_{d}$ for each $d$).

######  Lemma 3.22.

A conjugation-invariant short star-product $*$ is nondegenerate if and only if
the form $\langle a,\rho(b)\rangle$ is nondegenerate; in particular, a unitary
star-product is automatically nondegenerate.

###### Proof.

Straightforward. ∎

Also, recall that if $*$ is short and nondegenerate then it corresponds to a
nondegenerate $g$-twisted trace $T$ on some quantization ${\mathbf{A}}$ of
$A$, and vice versa. Moreover, if $A$ is equipped with a conjugation map
$\rho$ then we have an antilinear map $\rho\colon{\mathbf{A}}\to{\mathbf{A}}$
(as ${\mathbf{A}}$ is identified with $A$ as a filtered vector space).

######  Lemma 3.23.

  1. $(i)$

The star-product $*$ is conjugation-invariant if and only if $\rho$ is a
conjugation on ${\mathbf{A}}$ $($i.e., an antilinear algebra automorphism$)$
which conjugates $T$ $($hence commutes with $g)$;

  2. $(ii)$

In this situation $*$ is Hermitian if and only if in addition $\rho^{2}=g$.

###### Proof.

Straightforward. ∎

Let us now restrict to the case $A=\mathcal{O}(X)$ where $X$ is a conical
symplectic singularity.

######  Lemma 3.24.

A lift of a conjugation $\rho$ from $A$ to ${\mathbf{A}}$ is unique if exists.

###### Proof.

Any two lifts differ by an automorphism $\gamma$ of ${\mathbf{A}}$ that
commutes with $s$ and acts trivially on $A$, and any such automorphism is the
identity since any filtration-preserving derivation of ${\mathbf{A}}$ is inner
(this follows from Lemma 2.9 by taking the associated graded). ∎

Moreover, if $A$ is equipped with a conjugation then the space $\mathfrak{P}$
parametrizing quantizations of $A$ carries a real structure
$\lambda\mapsto\overline{\lambda}$ induced by $\rho$.

######  Lemma 3.25.

A quantization ${\mathbf{A}}_{\lambda}$ admits a lift of $\rho$ if and only if
$\overline{\lambda}\in W\lambda$, i.e., $\overline{\lambda}=\lambda$ in
$\mathfrak{P}/W$, where $W$ is the Namikawa Weyl group.

###### Proof.

Straightforward. ∎

###  3.7 Quaternionic structures

As before, let $A$ be a graded Poisson algebra with Poisson bracket of degree
$-2$.

######  Definition 3.26.

We say that a conjugation $\rho\colon A\to A$ is a quaternionic structure if
$\rho^{2}=s$.

If $\rho$ is a quaternionic structure then the operators $\rho$ and
$\sigma:={\rm i}^{d}$ define an (${\mathbb{R}}$-linear) action of the
quaternion group $Q_{8}$ by algebra automorphisms of $A$, which justifies the
terminology. Also it is clear that if $\rho$ is a quaternionic structure and
$\rho^{\prime}$ is equivalent to $\rho$ then $\rho^{\prime}$ is also a
quaternionic structure.

######  Definition 3.27.

Let ${\mathbf{A}}$ be an even quatization of $A$. We say that a conjugation
$\rho\colon{\mathbf{A}}\to{\mathbf{A}}$ is a quaternionic structure on
${\mathbf{A}}$ if $\rho^{2}=s$ and $\rho\circ\sigma=\sigma^{-1}\circ\rho$.

It is clear that any quaternionic structure on ${\mathbf{A}}$ gives rise to a
quaternionic structure on $A$ (by taking the associated graded), and a
quaternionic structure on $A$ uniquely lifts to a quantization
${\mathbf{A}}={\mathbf{A}}_{\lambda}$ in the case of conical symplectic
singularities when $\overline{\lambda}=\lambda=-\lambda\in\mathfrak{P}/W$.

Now suppose that $A$ is a Poisson algebra with a quaternionic structure $\rho$
and a short conjugation-invariant star-product $*$.

######  Lemma 3.28.

The star-product $*$ is Hermitian if and only if it is even.

###### Proof.

If $a,b\in A_{d}$ then

$\displaystyle
CT(\rho(a)*b)=\overline{CT(\rho(\rho(a)*b))}=\overline{CT(s(a)*\rho(b))}=(-1)^{d}\overline{CT(a*\rho(b))}.$

This implies the statement. ∎

Thus, conjugation-invariant Hermitian star-products on a quaternionic Poisson
algebra $A$ correspond to quaternionic quantizations ${\mathbf{A}}$ of $A$
together with a quantization map $\phi$ which commutes with $\sigma$ and
$\rho$.

###  3.8 The quarternionic structure on a HyperKähler cone

If $A=\mathcal{O}(X)$ where $X$ is a hyperKähler cone then $A$ has a canonical
quaternionic structure. Indeed, $X$ has a natural real analytic ${\rm
SU}(2)$-action (see [1, Appendix A and references therein]), so we have
degree-preserving operators $I$, $J$, $K$ on the space
${\mathcal{O}}(X_{{\mathbb{R}}})$ of real analytic polynomial functions on $X$
such that $IJ=JIs=K$ and $I^{2}=J^{2}=s$. The operator $I$ actually preserves
the subspace of holomorphic polynomials, $A={\mathcal{O}}(X)$, and acts on
$A_{d}$ by ${\rm i}^{d}$, i.e., $I=\sigma$. On the other hand, $J$ and $K$ map
holomorphic polynomials to antiholomorphic ones. Thus we have an antilinear
Poisson automorphism of $A$ given by $\rho(f)=\overline{Jf}$. Thus $\rho$
defines a quaternionic structure on $A$, which lifts uniquely to every
quantization ${\mathbf{A}}_{\lambda}$ of $A$ with
$\overline{\lambda}=\lambda=-\lambda\in\mathfrak{P}/W$.

######  Example 3.29.

The simplest example of a hyperKähler cone is a finite dimensional right
quaternionic vector space $V$ with a positive definite quaternionic Hermitian
inner product. Then $V$ carries a Euclidean metric $(\,,\,)$ given by the real
part of this inner product, and three real-linear operators $I$, $J$, $K$
corresponding to three complex structures on $V$, which gives a hyperKähler
structure on $V$. Let $A$ be the algebra of polynomial functions on the
complex vector space $(V,I)$. We have three real symplectic forms on $V$,

$\displaystyle\omega_{I}(v,w)=(Iv,w),\qquad\omega_{J}(v,w)=(Jv,w),\qquad\omega_{K}(v,w)=(Kv,w).$

The form $\omega=\omega_{J}+{\rm i}\omega_{K}$ is holomorphic for the complex
structure $I$, so defines a Poisson bracket on $A$ of degree $-2$, whose
symmetry group is ${\rm Sp}(V)$. The symmetry group of $((\,,\,),I,J,K)$ is a
maximal compact subgroup ${\rm U}(V)$ of ${\rm Sp}(V)$, the quaternionic
unitary group. As explained above, we have $\sigma(f)(v)=f({\rm i}v)$,
$\rho(f)(v)=\overline{f(vJ)}$.

Now let $V$ be a complex vector space with a symplectic form $\omega$ and
$G\subset{\rm Sp}(V)$ be a finite subgroup. There exists a positive definite
quaternionic Hermitian structure on $V$ such that $G\subset{\rm U}(V)$, which
is unique up to a positive scalar if $V$ is indecomposable as a symplectic
representation of $G$. A choice of such a structure gives us a form $(\,,\,)$
and operators $I$, $J$, $K$ preserved by $G$, so $X=V/G$ is also a hyperKähler
cone, and $A:=\mathcal{O}(X)$ is a quaternionic Poisson algebra.

Let us now classify quaternionic structures on a hyperKähler cone $X$. As
explained above, there is a canonical one, call it $\rho_{0}$. Note that
$\operatorname{Ad}\rho_{0}$ is an involution on
$\operatorname{Aut}({\mathbf{A}})$ which corresponds to its compact real form,
denote it by $\gamma\mapsto\overline{\gamma}$. Now, any quaternionic structure
$\rho$ is of the form $\rho=\rho_{0}\gamma$ for some automorphism
$\gamma\in\operatorname{Aut}({\mathbf{A}})$. The condition on $\gamma$ is that
$s=\rho^{2}=\rho_{0}\gamma\rho_{0}\gamma=s\overline{\gamma}\gamma$, i.e.,
$\overline{\gamma}\gamma=1$. Moreover, replacing $\rho$ by an equivalent
conjugation $h\rho h^{-1}$ corresponds to replacing $\gamma$ with
$\overline{h}\gamma h^{-1}$. This implies that equivalence classes of
quaternionic structures correspond to the real forms of
$\operatorname{Aut}({\mathbf{A}})$ given by the Galois cohomology classes in
$H^{1}({\mathbb{Z}}/2,\operatorname{Aut}({\mathbf{A}}))$, where
${\mathbb{Z}}/2$ acts by $\gamma\mapsto\overline{\gamma}$; i.e., to real forms
which are in the inner class of the compact form.101010Note that the group
$\operatorname{Aut}({\mathbf{A}})$ may be disconnected.

######  Example 3.30.

If $X$ is the nilpotent cone of a semisimple Lie algebra $\mathfrak{g}$ (so
$s=1$) then ${\mathbf{A}}={\mathbf{A}}_{\lambda}$ is a maximal quotient of the
enveloping algebra ${\rm U}(\mathfrak{g})$ by the central character
$\chi=\chi_{\lambda}$ of $\lambda$, and for conjugations to exist, this
central character should be real. Then $\operatorname{Aut}({\mathbf{A}})$ is
the subgroup $\operatorname{Aut}(\mathfrak{g})_{\chi}$ of
$\operatorname{Aut}(\mathfrak{g})$ that preserves $\chi$. So the real forms we
get are those which differ from the compact form by an involution
$\tau\in\operatorname{Aut}(\mathfrak{g})_{\chi}$ (i.e., this form is defined
by the equation $\overline{g}=\tau(g)$). In particular, if the automorphism
group of the Dynkin diagram of $\mathfrak{g}$ preserves $\chi$ (e.g., if this
group is trivial) then conjugations up to equivalence correspond to real forms
of $\mathfrak{g}$. So for $\mathfrak{g}=\mathfrak{sl}_{2}$ we have two
conjugations up to equivalence – the compact one and the noncompact one.

###  3.9 Classification of Hermitian short star-products

Let $X$ be a conical symplectic singularity equipped with a conjugation
$\rho\colon A\to A$ of $A=\mathcal{O}(X)$ (for example, a hyperKähler cone,
cf. Section 3.8). In this case, given $\lambda\in\mathfrak{P}$ with
$\overline{\lambda}\in W\lambda$, we have the corresponding unique lift
$\rho\colon{\mathbf{A}}_{\lambda}\to{\mathbf{A}}_{\lambda}$. Let $g=\rho^{2}$.

Recall that by Corollary 3.5 nondegenerate short star-products for
${\mathbf{A}}_{\lambda}$ corresponding to $g$ are classified by nondegenerate
traces $T\in(HH_{0}({\mathbf{A}}_{\lambda},{\mathbf{A}}_{\lambda}g)^{*})^{s}$
such that $T(1)=1$. Note that $g$ acts trivially on
$(HH_{0}({\mathbf{A}}_{\lambda},{\mathbf{A}}_{\lambda}g)^{*})^{s}$, hence
$\rho$ defines an antilinear involution $\rho_{*}$ (i.e., a real structure) on
this vector space. Therefore, by Lemma 3.23 we have the following proposition.

######  Proposition 3.31.

With the above assumptions, Hermitian nondegenerate short star-products for
${\mathbf{A}}_{\lambda}$ correspond to real nondegenerate traces
$T\in(HH_{0}({\mathbf{A}}_{\lambda},{\mathbf{A}}_{\lambda}g)^{*})^{s}$ such
that $T(1)=1$, i.e., those invariant under $\rho_{*}$.

In particular, nondegenerate short star-products corresponding to $g$ exist if
and only if there exist Hermitian ones (and in this case a Weil generic short
star-product is nondegenerate). In other words, the Hermitian property “goes
along for the ride”.

######  Example 3.32.

Consider again the case when $X$ is the quadratic cone in ${\mathbb{C}}^{3}$,
i.e., the $\mathfrak{sl}_{2}$ case. Recall that a nondegenerate short star-
product exists if and only if the central character
$\chi=\chi_{\lambda}:=\lambda(\lambda+2)/2$ is not equal to $n(n+2)/2$ where
$n\geq 0$ is an integer. Thus, for each such $\chi\in{\mathbb{R}}$ and each
conjugation $\rho$ we have a unique Hermitian even nondegenerate short star-
product (the one invariant under $\mathfrak{sl}_{2}$). One can show (see,
e.g., [1]) that for the compact conjugation, this product is never unitary. On
the other case, a direct computation shows that for the non-compact
conjugation, this product is unitary if and only if $\chi<0$ (see [1, Section
5.1]). This is related to Bargmann’s classification of unitary representations
of ${\rm SL}_{2}({\mathbb{R}})$ (the principal and complementary series, which
exist in this range).

Thus, for the nilpotent cone of higher rank simple Lie algebras (as well as
orbit closures and Slodowy slices) we also expect a relation of the question
of unitarity of short star-products and the theory of unitary representations
of real forms of the Lie group $G$ corresponding to $\mathfrak{g}$. Hopefully
this will be a first step in the general theory of unitary representations of
quantizations of symplectic singularities, which would generalize the theory
of unitary representations of real reductive groups. We plan to discuss this
in more detail in the second part of this work.

##  4 Existence of nondegenerate short star-products

###  4.1 The case $\boldsymbol{g=s}$

Let us now discuss existence of nondegenerate short star-products. We first
focus on the case $g=s$, which includes the case of even star-products. We
have shown that such star-products giving a quantization ${\mathbf{A}}$ are
parametrized by a subset of the space
$(HH_{0}({\mathbf{A}},{\mathbf{A}}s)^{s})^{*}$ determined by a nondegeneracy
condition.

######  Conjecture 4.1.

  1. $(i)$

For Weil generic $\lambda$ a Weil generic element
$T\in(HH_{0}({\mathbf{A}}_{\lambda},{\mathbf{A}}_{\lambda}s)^{s})^{*}$ is
nondegenerate, so it defines a nondegenerate short star-product corresponding
to $g=s$.

  2. $(ii)$

For Weil generic $\lambda$ such that $-\lambda\in W\lambda$ a Weil generic
element
$T\in(HH_{0}({\mathbf{A}}_{\lambda},{\mathbf{A}}_{\lambda}s)^{\sigma})^{*}$ is
nondegenerate, so it defines an even nondegenerate short star-product.

This conjecture generalizes a conjecture from [1].

Below we check Conjecture 4.1 in a number of important special cases.

####  4.1.1 Spherical symplectic reflection algebras

Let $V$ be a finite dimensional symplectic vector space with symplectic form
$\omega$, and $G\subset{\rm Sp}(V)$ a finite subgroup such that
$\big{(}{\wedge}^{2}V^{*}\big{)}^{G}={\mathbb{C}}\omega$. Let $S$ be the set
of symplectic reflections in $G$ (i.e., elements $g$ such that $(1-g)|_{V}$
has rank $2$) and $c\colon S\to{\mathbb{C}}$ a $G$-invariant function. The
symplectic reflection algebra ${\mathbf{H}}_{c}={\mathbf{H}}_{c}(G,V)$
attached to this data is the quotient of ${\mathbb{C}}G\ltimes TV$ by the
relations

$\displaystyle[v,w]=\omega(v,w)+\sum_{g\in S}c_{g}\omega((1-g)v,(1-g)w)g,$

see [10]. Let ${\mathbf{e}}=|G|^{-1}\sum\limits_{g\in G}g$. The spherical
subalgebra of ${\mathbf{H}}_{c}$ is the algebra
${\mathbf{A}}_{c}:={\mathbf{e}}{\mathbf{H}}_{c}{\mathbf{e}}$. It is shown in
[10] that ${\mathbf{A}}_{c}$ is a filtered deformation of $A:=(SV)^{G}$.
Moreover, by the result of [19], this is the most general such deformation (as
in this case $\mathfrak{P}\cong{\mathbb{C}}[S]^{G}$); here $\lambda$ is
related to $c$ by a nonhomogeneous invertible linear transformation.

######  Theorem 4.2.

Conjecture 4.1 holds for ${\mathbf{A}}_{c}$.

###### Proof.

Recall that the automorphism $s$ is defined by $s|_{V}=-1$. Let us compute the
space $HH_{0}({\mathbf{A}}_{c},{\mathbf{A}}_{c}s)^{s}$ for Weil generic $c$.
By the results of [10], the algebra ${\mathbf{A}}_{c}$ is Morita equivalent to
${\mathbf{H}}_{c}$, so this space is isomorphic to
$HH_{0}({\mathbf{H}}_{c},{\mathbf{H}}_{c}s)^{s}$ (as this Morita equivalence
is $s$-invariant).

Consider first the case when $s\in G$. In this case,
$HH_{0}({\mathbf{H}}_{c},{\mathbf{H}}_{c}s)^{s}=HH_{0}({\mathbf{H}}_{c},{\mathbf{H}}_{c})$.
This group was computed in [10] and is isomorphic to
${\mathbb{C}}[\Sigma_{+}]^{G}$, where $\Sigma_{+}$ is the set of elements of
$G$ such that $(1-g)|_{V}$ is invertible. Note also that the same result holds
for $c=0$.

Now consider the case $s\notin G$. In this case a similar argument shows that
$HH_{0}({\mathbf{H}}_{c},{\mathbf{H}}_{c}s)^{s}$ is isomorphic to
${\mathbb{C}}[\Sigma_{-}]^{G}$, where $\Sigma_{-}$ is the set of elements of
$G$ such that $(1+g)|_{V}$ is invertible, and again this result holds for
$c=0$. Note that this holds regardless of whether $s$ belongs to $G$, since
when $s\in G$ then multiplication by $s$ defines a bijection
$\Sigma_{+}\to\Sigma_{-}$. Note also that this space is always nonzero since
$1\in\Sigma_{-}$.

Finally, ${\mathbf{A}}_{0}={\rm Weyl}(V)^{G}$, and this algebra admits a
nondegenerate $s$-twisted trace corresponding to the Moyal–Weyl product. By a
deformation argument, this implies that for Weil generic $c\neq 0$, a Weil
generic element of ${\mathbb{C}}[\Sigma_{-}]^{G}$ gives rise to a
nondegenerate $s$-twisted trace, as desired.

Finally, in the even case we have to show that we have a nondegenerate
$\sigma$-invariant trace. It is easy to see that for $c=0$ the map $\sigma$
acts on ${\mathbb{C}}[\Sigma_{-}]^{G}$ by $\sigma(g)=g^{-1}$. This shows that
even nondegenerate star-products exist for Weil generic $c$ and are
parametrized by Weil generic $G$-invariant functions on $\Sigma_{-}$ stable
under $g\mapsto g^{-1}$, modulo scaling. ∎

######  Example 4.3.

Suppose $\dim V=2$. This corresponds to the case of Kleinian singularities,
and the possible groups $G$ are in bijection with simply-laced finite Dynkin
diagrams via the McKay’s correspondence. Namely, $G={\mathbb{Z}}/m$ for
$A_{m-1}$, $G$ is the double cover of the dihedral group $I_{m-2}$ for
$D_{m}$, of the tetrahedral group ${\mathbb{A}}_{4}$ for $E_{6}$, the cube
group $S_{4}$ for $E_{7}$ and the icosahedral group ${\mathbb{A}}_{5}$ for
$E_{8}$. The Namikawa Weyl group coincides with the usual Weyl group of the
corresponding root system. Looking at the conjugacy classes of elements not
equal to $-1$ modulo $g\mapsto g^{-1}$, we find the following facts about even
nondegenerate short star-products.

  1. 1.

The number $m_{e}$ of parameters for even quantizations equals the number of
orbits of the dualization involution (for Lie algebra representations) on the
finite Dynkin diagram. So $m_{e}$ equals
$\big{\lfloor}\frac{m+1}{2}\big{\rfloor}$ for $A_{m}$, $m-1$ for $D_{m}$ with
$m$ odd, $4$ for $E_{6}$, and $m$ in all other cases.

  2. 2.

The number of parameters for even nondegenerate short star-products for each
quantization is $m_{e}-1$, except the case of $A_{m}$ with even $m$, when it
equals $m_{e}$.

####  4.1.2 Maximal quotients of the enveloping algebra of a simple Lie
algebra

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra with Cartan
subalegbra $\mathfrak{h}$, ${\rm U}(\mathfrak{g})$ its enveloping algebra,
$Z(\mathfrak{g})$ the center of ${\rm U}(\mathfrak{g})$,
$\lambda\in\mathfrak{h}^{*}$ a complex weight for $\mathfrak{g}$, and
$\chi\colon Z(\mathfrak{g})\to{\mathbb{C}}$ the central character of the Verma
module with highest weight $\lambda$. Let
${\mathbf{A}}_{\lambda}:=U_{\chi}(\mathfrak{g})={\rm
U}(\mathfrak{g})/(z-\chi(z),z\in Z(\mathfrak{g}))$ be the corresponding
maximal primitive quotient. Then ${\mathbf{A}}_{\lambda}$ is naturally a
filtered quantization of $A=\mathcal{O}(X)$, where $X$ is the nilpotent cone
of $\mathfrak{g}$ (where elements of $\mathfrak{g}$ have degree $2$). In this
case, since all the degrees are even, we have $s=1$. Moreover, if $\chi$ is
self-dual, the antipode of ${\rm U}(\mathfrak{g})$ defines the
antiautomorphism $\sigma$ of ${\mathbf{A}}_{\lambda}$, so that
${\mathbf{A}}_{\lambda}$ is an even quantization of $A$.

Let us first classify nondegenerate untwisted traces. It is well known that
$HH_{0}({\mathbf{A}}_{\lambda},{\mathbf{A}}_{\lambda})={\mathbb{C}}$. Namely,
if $\chi$ is the central character of a finite dimensional representation $V$
then the unique trace $T$ normalized by the condition $T(1)=1$ is given by the
normalized character of $V$:

$\displaystyle T(a)=\frac{\operatorname{Tr}(a|_{V})}{\dim V}.$

This trace is defined by a rational function of $\chi$, so for general $\chi$
it can be obtained by interpolation.

######  Proposition 4.4.

$T$ is nondegenerate for Weil generic $\lambda$ $($or $\chi)$.

###### Proof.

It suffices to show that for each $i$, $T$ is nondegenerate on
$F_{i}{\mathbf{A}}_{\lambda}$ for Weil generic $\lambda$. To this end, it
suffices to show that this is so for at least one $\lambda$. To do so, take
$\lambda$ dominant integral and consider the irreducible finite dimensional
representation $V=V_{\lambda}$ of $\mathfrak{g}$. We claim that for each $i$
one can pick $\lambda$ so that $F_{i}{\mathbf{A}}_{\lambda}$ has trivial
intersection with the annihilator $\operatorname{Ann}(V)$ of $V$ in
${\mathbf{A}}_{\lambda}$. Indeed, recall that as a $\mathfrak{g}$-module,
${\mathbf{A}}_{\lambda}=\oplus_{\mu}V_{\mu}\otimes V_{\mu}^{*}[0]$, where
$\mu$ runs over dominant integral weights. On the other hand,
$\operatorname{End}(V)=\oplus_{\mu}V_{\mu}\otimes\operatorname{Hom}(V_{\mu}\otimes
V,V)$. So if $\operatorname{Hom}(V_{\mu}\otimes V,V)\cong V_{\mu}^{*}[0]$ for
all $V_{\mu}$ occurring in $F_{i}{\mathbf{A}}_{\lambda}$, then
$F_{i}{\mathbf{A}}_{\lambda}\cap\operatorname{Ann}(V)=0$. Clearly, this
happens if all the coordinates of $\lambda$ are large enough.

Now note that $V$ is a unitary representation, so $\operatorname{End}(V)$ has
an antilinear antiautomorphism $B\mapsto B^{\dagger}$, such that
$\operatorname{Tr}_{V}\big{(}B^{\dagger}B\big{)}>0$ for $B\neq 0$. Let
$\pi_{V}\colon{\mathbf{A}}_{\lambda}\to\operatorname{End}(V)$ be the
representation map. We have
$\pi_{V}({\mathbf{a}})^{\dagger}=\pi_{V}\big{(}{\mathbf{a}}^{\dagger}\big{)}$,
where ${\mathbf{a}}\mapsto{\mathbf{a}}^{\dagger}$ is the compact $*$-structure
on ${\mathbf{A}}_{\lambda}$. For ${\mathbf{a}}\in{\mathbf{A}}_{\lambda}$, we
have

$\displaystyle\big{(}{\mathbf{a}},{\mathbf{a}}^{\dagger}\big{)}_{T}=\frac{\operatorname{Tr}_{V}\big{(}\pi_{V}({\mathbf{a}})\pi_{V}({\mathbf{a}})^{\dagger}\big{)}}{\dim
V}.$

If ${\mathbf{a}}\in F_{i}{\mathbf{A}}_{\lambda}$ then
$\pi_{V}({\mathbf{a}})\neq 0$, so we get that
$\big{(}{\mathbf{a}},{\mathbf{a}}^{\dagger}\big{)}_{T}>0$. Thus $(\,,\,)_{T}$
is nondegenerate on $F_{i}{\mathbf{A}}_{\lambda}$, as desired. ∎

######  Remark 4.5.

More generally, let $y\in\mathfrak{h}\subset\mathfrak{g}$ be a semisimple
element, $L=Z_{y}$ its centralizer (a Levi subgroup of the corresponding group
$G$) and $P$ a parabolic in $G$ whose Levi subgroup is $L$. Let $U$ be the
unipotent radical of $P$ and $u\in\operatorname{Lie}U\subset\mathfrak{g}$ a
Richardson element (i.e., one whose $P$-orbit is dense in
$\operatorname{Lie}U$), which exists by a classical theorem of Richardson. We
view $u$ as an element of $\mathfrak{g}^{*}$ and consider the coadjoint orbit
$\mathcal{O}_{u}$ of $u$, called the Richardson orbit corresponding to $y$
(e.g., for $\mathfrak{g}=\mathfrak{sl}_{n}$ any nilpotent orbit arises in this
way). Let $X$ be the closure of $\mathcal{O}_{u}$. Let $J\subset I$ be the set
of $j$ such that $\alpha_{j}(y)=0$ (e.g., if $X$ is the nilpotent cone then
$y$ is regular and $J=\varnothing$ and if $X=0$ then $y=0$ and $J=I$). In this
case $\mathcal{O}(X)$ admits a family of quantizations
${\mathbf{A}}_{\lambda}$ parametrized by weights $\lambda$ such that
$\lambda(h_{i})=0$ for $i\in J$; namely, ${\mathbf{A}}_{\lambda}$ is the image
of ${\rm U}(\mathfrak{g})$ in the ${\mathbb{C}}$-endomorphism algebra of the
parabolic Verma module $M_{P}(\lambda)$ with highest weight $\lambda$
(generated by a vector $v$ with defining relations $e_{i}v=0$,
$h_{i}v=\lambda(h_{i})v$ for all $i$ and $f_{i}v=0$ for $i\in J$). In this
case, we still have $s=1$ and
$HH_{0}({\mathbf{A}}_{\lambda},{\mathbf{A}}_{\lambda})={\mathbb{C}}$, so we
have a unique trace up to scaling. A similar positivity argument to the above,
using finite dimensional quotients of $M_{P}(\lambda)$ (which occur when
$\lambda$ is dominant integral), shows that this trace is nondegenerate for
Weil generic $\lambda$, hence gives rise to a nondegenerate even short star-
product.

###  4.2 The general case

Finally, let us describe the construction of nondegenerate twisted traces for
generic $g\in\operatorname{Aut}({\mathbf{A}})^{0}$, in the case of hyperKähler
cones. We plan to give more details and also treat the case when
$g\notin\operatorname{Aut}({\mathbf{A}})^{0}$ in a subsequent paper.

####  4.2.1 Characters of Verma modules

We may assume that $g$ is in a maximal torus
${\mathbf{T}}\subset\operatorname{Aut}({\mathbf{A}})$. Let us adapt the
construction from Sections 3.9, 3.10 which uses representation theory
(category $\mathcal{O}$). Namely, let
$\nu\colon{\mathbb{C}}^{\times}\to{\mathbf{T}}$ be a 1-parameter subgroup. By
Lemma 2.9, the derivation defined by $\nu$ is inner and given by some element
${\mathbf{h}}\in{\mathbf{A}}$, such that the operator $[{\mathbf{h}},?]$ on
${\mathbf{A}}$ is diagonalizable with integer spectrum. For $m\in{\mathbb{Z}}$
denote by ${\mathbf{A}}(m)$ the space of ${\mathbf{a}}\in{\mathbf{A}}$ such
that $[{\mathbf{h}},{\mathbf{a}}]=m{\mathbf{a}}$; we have
${\mathbf{A}}=\oplus_{m\in{\mathbb{Z}}}{\mathbf{A}}(m)$.

Following [21, Section 4] and [3] (see also [13]), in this situation one can
define the category $\mathcal{O}=\mathcal{O}(\nu,\lambda)$ of representations
of ${\mathbf{A}}={\mathbf{A}}_{\lambda}$. Namely, we define

$\displaystyle{\mathbf{A}}^{>0}:=\oplus_{m>0}{\mathbf{A}}(m),\qquad{\mathbf{A}}^{<0}:=\oplus_{m<0}{\mathbf{A}}(m),$
$\displaystyle{\mathbf{A}}^{\geq 0}:=\oplus_{m\geq
0}{\mathbf{A}}(m),\qquad{\mathbf{A}}^{\leq 0}:=\oplus_{m\leq
0}{\mathbf{A}}(m).$

Also we define the algebra

$\displaystyle
C_{\nu}({\mathbf{A}})={\mathbf{A}}(0)/\sum_{m>0}{\mathbf{A}}(-m){\mathbf{A}}(m),$

called the Cartan subquotient of ${\mathbf{A}}$. We define the category
$\mathcal{O}(\nu,\lambda)$ to be the category of finitely generated
${\mathbf{A}}$-modules $M$ on which ${\mathbf{A}}^{>0}$ acts locally
nilpotently (i.e., for every vector $v\in M$ the space ${\mathbf{A}}^{>0}v$ is
finite dimensional and has a filtration strictly preserved by
${\mathbf{A}}^{>0}$).

Let us assume that the only fixed point of ${\mathbf{T}}$ on $X$ is the
origin. In this case we can (and will) choose $\nu$ to be so generic that
$X^{\nu({\mathbb{C}}^{\times})}=X^{{\mathbf{T}}}$ as schemes. Then the algebra
$C_{\nu}({\mathbf{A}})$ is finite dimensional and carries a trivial action of
${\mathbf{T}}$. In this case, it is shown in [21, Lemma 4.1] that the category
$\mathcal{O}$ is artinian with finitely many simple objects and enough
projectives and each object has finite dimensional generalized eigenspaces of
${\mathbf{h}}$. Also $\mathcal{O}$ has an important class of objects called
Verma modules. Namely, given an (irreducible) $C_{\nu}({\mathbf{A}})$-module
$\tau$, the Verma module $\Delta(\tau)$ with lowest weight $\tau$ is, by
definition, generated by a copy of $\tau$ (regarded as an
${\mathbf{A}}(0)$-module) with defining relations $av=0$ for
$a\in{\mathbf{A}}^{<0}$, $v\in\tau$.

Let us equip $\tau$ with the trivial action of ${\mathbf{T}}$. This defines an
action of ${\mathbf{T}}$ on $\Delta(\tau)$ compatible with the
${\mathbf{A}}$-action. Define the reduced character of $\Delta(\tau)$ by the
formula

$\displaystyle{\rm
Ch}_{\tau}({\mathbf{a}},g,t)=\operatorname{Tr}_{\Delta(\tau)}({\mathbf{a}}g\nu(t)):=\sum_{m\geq
0}\operatorname{Tr}_{\Delta(\tau)[m]}({\mathbf{a}}g)t^{m}\in{\mathbb{C}}[[t]],$

where ${\mathbf{a}}\in{\mathbf{A}}(0)$ and $g\in{\mathbf{T}}$. By looking at
the constant terms of these series, it is clear that these characters for
different irreducible representations $\tau$ are linearly independent over
${\mathbb{C}}((t))$ (for each $g\in{\mathbf{T}}$).

It is clear that ${\rm Ch}_{\tau}(?,g,t)$ is a $g\nu(t)$-twisted trace on the
algebra ${\mathbf{A}}^{t}:={\mathbf{A}}\otimes{\mathbb{C}}((t))$. Thus we
obtain

######  Proposition 4.6.

We have a ${\mathbb{C}}((t))$-linear injection

$\displaystyle{\rm Ch}\colon\ K_{0}(C_{\nu}({\mathbf{A}}))((t))\hookrightarrow
HH_{0}\big{(}{\mathbf{A}}^{t},{\mathbf{A}}^{t}g\nu(t)\big{)}^{*}.$

####  4.2.2 Differential equations for reduced characters

Let ${\mathbf{b}}_{i}$, $i=1,\dots,r$ be a collection of elements of
${\mathbf{A}}(0)$ such that any element ${\mathbf{a}}\in F_{k}{\mathbf{A}}(0)$
can be written as

$\displaystyle{\mathbf{a}}=\sum_{i}\lambda_{i}{\mathbf{b}}_{i}+\sum_{j}{\mathbf{c}}_{j}^{-}{\mathbf{c}}_{j}^{+},\qquad\lambda_{i}\in{\mathbb{C}},$

where ${\mathbf{c}}_{j}^{\pm}\in{\mathbf{A}}(\pm m_{j})$ for some $m_{j}>0$,
so that $\deg({\mathbf{c}}_{j}^{+})+\deg({\mathbf{c}}_{j}^{-})\leq k$. Such a
collection exists since the only fixed point of $\nu({\mathbb{C}}^{\times})$
on $X$ is the origin. Moreover, we may assume that ${\mathbf{c}}_{j}^{\pm}$
have weights $\pm\mu_{j}$ under ${\mathbf{T}}$.

Let $T$ be a $g\nu(t)$-twisted trace on ${\mathbf{A}}^{t}$. Then we get

$\displaystyle
T({\mathbf{a}})=\sum_{i}\lambda_{i}T({\mathbf{b}}_{i})+\sum_{j}T({\mathbf{c}}_{j}^{-}{\mathbf{c}}_{j}^{+}).$

On the other hand,

$\displaystyle
T({\mathbf{c}}_{j}^{-}{\mathbf{c}}_{j}^{+})=T([{\mathbf{c}}_{j}^{-},{\mathbf{c}}_{j}^{+}])+T({\mathbf{c}}_{j}^{+}{\mathbf{c}}_{j}^{-})=T([{\mathbf{c}}_{j}^{-},{\mathbf{c}}_{j}^{+}])+\mu_{j}(g\nu(t))T({\mathbf{c}}_{j}^{-}{\mathbf{c}}_{j}^{+}),$

which implies that

$\displaystyle
T({\mathbf{c}}_{j}^{-}{\mathbf{c}}_{j}^{+})=\frac{T([{\mathbf{c}}_{j}^{-},{\mathbf{c}}_{j}^{+}])}{1-\mu_{j}(g\nu(t))}.$

Thus we get

$\displaystyle
T({\mathbf{a}})=\sum_{i}\lambda_{i}T({\mathbf{b}}_{i})+\sum_{j}\frac{T([{\mathbf{c}}_{j}^{-},{\mathbf{c}}_{j}^{+}])}{1-\mu_{j}(g\nu(t))}.$

But the element $[{\mathbf{c}}_{j}^{-},{\mathbf{c}}_{j}^{+}]$ has filtration
degree $\leq k-2$. Thus, continuing this process, we obtain the following
lemma.

Let $R\subset{\mathbb{C}}({\mathbf{T}})$ be the subalgebra generated by
elements of the form $\frac{1}{1-\mu}$, where $\mu$ is a character of
${\mathbf{T}}$ occurring in ${\mathbf{A}}(m)$ for some $m>0$.

######  Lemma 4.7.

For any ${\mathbf{a}}\in{\mathbf{A}}$ there exist $p_{i}\in R$, $i=1,\dots,r$
$($independent of $T)$ such that

$\displaystyle T({\mathbf{a}})=\sum_{i}p_{i}(g\nu(t))T({\mathbf{b}}_{i}).$

######  Corollary 4.8.

Let $h_{\tau}$ be the eigenvalue of ${\mathbf{h}}$ on
$\tau\subset\Delta(\tau)$. There exist $p_{ij}\in R$ independent of $\tau$
such that the power series ${\rm Ch}_{\tau}({\mathbf{b}}_{i},g,t)$ satisfy the
system of linear differential equations

$\displaystyle t\frac{{\rm d}}{{\rm d}t}{\rm
Ch}_{\tau}({\mathbf{b}}_{i},g,t)=\sum_{j=1}^{r}\big{(}p_{ij}(g\nu(t))-h_{\tau}\delta_{ij}\big{)}{\rm
Ch}_{\tau}({\mathbf{b}}_{j},g,t).$

In particular, these power series converge in some neighborhood of $0$ and
extend to $($a priori, possibly multivalued$)$ analytic functions outside
finitely many points $t={\rm e}^{2\pi ij/m}\mu(g)^{1/m}$ where $\mu$ is a
${\mathbf{T}}$-weight occurring in ${\mathbf{A}}(m)$, $m>0$.

###### Proof.

We have

$\displaystyle t\frac{{\rm d}}{{\rm d}t}{\rm
Ch}_{\tau}({\mathbf{b}}_{i},g,t)={\rm
Ch}_{\tau}(({\mathbf{h}}-h_{\tau}){\mathbf{b}}_{i},g,t).$

By Lemma 4.7, there exist $p_{ij}\in R$ such that

$\displaystyle{\rm
Ch}_{\tau}({\mathbf{h}}{\mathbf{b}}_{i},g,t)=\sum_{j}p_{ij}(g\nu(t)){\rm
Ch}_{\tau}({\mathbf{b}}_{j},g,t).$

This implies the statement. ∎

In particular, for Weil generic $g$ the reduced character ${\rm
Ch}_{\tau}({\mathbf{b}}_{i},g,t)$ can be evaluated at $t=1$, giving a
$g$-twisted trace on ${\mathbf{A}}$. Thus we obtain an injective map

$\displaystyle{\rm Ch}|_{t=1}\colon\
K_{0}(C_{\nu}({\mathbf{A}}))\hookrightarrow
HH_{0}({\mathbf{A}},{\mathbf{A}}g)^{*}.$

for Weil generic $g\in{\mathbf{T}}$. We will see that this map is often an
isomorphism, in which case we get a construction of all $g$-twisted traces via
representation theory.

####  4.2.3 Rationality of reduced characters

######  Theorem 4.9.

The series ${\rm Ch}_{\tau}({\mathbf{a}},g,t)$ converges to a rational
function of $g\nu(t)$ with numerator in ${\mathbb{C}}[{\mathbf{T}}]$ and
denominator a finite product of binomials of the form $1-\mu$, where $\mu$ is
a character of ${\mathbf{T}}$ occurring in ${\mathbf{A}}(m)$ for some $m>0$.

###### Proof.

Recall first the classical theorem of Carlson and Polya [5, 28].

######  Theorem 4.10.

Let $f(z)=\sum\limits_{n=0}^{\infty}c_{n}z^{n}\in{\mathbb{Z}}[[z]]$. If this
series converges for $|z|<1$ and analytically continues to any larger open set
then $f$ is rational.

Theorem 4.10 immediately implies

######  Corollary 4.11.

Let $f(z)=\sum\limits_{n=0}^{\infty}c_{n}z^{n}\in{\mathbb{Z}}[[z]]^{m}$.
Suppose that $f$ satisfies a linear ODE of the form

$\displaystyle f^{\prime}(z)=A(z)f(z),$

where $A(z)\in\operatorname{Mat}_{m}({\mathbb{Q}}(z))$ is regular for $|z|<1$.
Then $f$ is rational.

Let us now use Corollary 4.11 to prove the rationality of ${\rm
Ch}_{\tau}({\mathbf{a}},g,t)$ for $g$ of finite order $r$. Clearly, it is
enough to check the rationality of the series

$\displaystyle f_{j,\tau}({\mathbf{a}},g,t):=\sum_{l=0}^{r-1}{\rm e}^{2\pi
ijl}{\rm Ch}_{\tau}\big{(}{\mathbf{a}},g^{l},t\big{)}$

for all $j\in[0,r-1]$.

By standard abstract nonsense, it suffices to consider the case when the
ground field is $\overline{{\mathbb{Q}}}$. Moreover, since ${\mathbf{A}}$ is
finitely generated, we may assume that the ground field is some (Galois)
number field $K$.

Furthermore, restricting scalars, we may (and will) assume that
$K={\mathbb{Q}}$. Indeed, let ${\mathbf{A}}_{{\mathbb{Q}}}$ be the algebra
${\mathbf{A}}$ regarded as a ${\mathbb{Q}}$-algebra. Let $\Gamma={\rm
Gal}(K/{\mathbb{Q}})$. It is clear that

$\displaystyle{\mathbf{A}}_{{\mathbb{Q}}}\otimes
K=\oplus_{\gamma\in\Gamma}{\mathbf{A}}^{\gamma},$ (4.1)

where ${\mathbf{A}}^{\gamma}$ is ${\mathbf{A}}$ twisted by $\gamma$. Let
$f_{j,\tau}^{{\mathbb{Q}}}$ be the series $f_{j,\tau}$ computed for the
algebra ${\mathbf{A}}_{{\mathbb{Q}}}$, i.e., over ${\mathbb{Q}}$. By (4.1), we
have

$\displaystyle
f_{j,\tau}^{{\mathbb{Q}}}({\mathbf{a}},g,t)=\sum_{\gamma\in\Gamma}\gamma(f_{j,\tau}({\mathbf{a}},g,t))=\operatorname{Tr}_{K/{\mathbb{Q}}}(f_{j,\tau}({\mathbf{a}},g,t)).$

Now let $c_{i}$ be a basis of $K$ over ${\mathbb{Q}}$ and $c_{i}^{*}$ the dual
basis with respect to the form $\operatorname{Tr}_{K/{\mathbb{Q}}}(xy)$. Then

$\displaystyle
f_{j,\tau}({\mathbf{a}},g,t)=\sum_{i}c_{i}^{*}\operatorname{Tr}_{K/{\mathbb{Q}}}\big{(}c_{i}f_{j,\tau}({\mathbf{a}},g,t)\big{)}=\sum_{i}c_{i}^{*}f_{j,\tau}^{{\mathbb{Q}}}(c_{i}{\mathbf{a}},g,t).$

Thus, it suffices to prove the statement for ${\mathbf{A}}_{{\mathbb{Q}}}$, as
claimed.

Using Lemma 4.7, Corollary 4.8, and Corollary 4.11, we see that rationality of
$f_{j,\tau}({\mathbf{a}},g,t)$ follows from the following proposition, which
shows that an integer multiple of $f_{j,\tau}({\mathbf{a}},g,t)$ has integer
coefficients.

######  Proposition 4.12.

There exists a basis $B$ of $\Delta(\tau)$ compatible with the grading such
that for each ${\mathbf{a}}\in{\mathbf{A}}(0)$ there exists an integer
$N_{{\mathbf{a}}}\geq 1$ for which
${\mathbf{a}}B\subset\frac{1}{N_{{\mathbf{a}}}}{\mathbb{Z}}B$.

###### Proof.

We have $\Delta(\tau)={\mathbf{A}}^{\leq 0}\tau$. Recall that the algebra
${\mathbf{A}}^{\leq 0}$ is finitely generated. Let $x_{1},\dots,x_{r}$ be
homogeneous generators of ${\mathbf{A}}^{\leq 0}$, where $x_{i}$ have negative
degree for $i\leq s$ and degree $0$ for $i>s$. Then $x_{s+1},\dots,x_{r}$
generate ${\mathbf{A}}(0)$. So it suffices to show that $N_{{\mathbf{a}}}$
exists for ${\mathbf{a}}=x_{i}$, $i>s$, i.e., that $x_{s+1},\dots,x_{r}$ can
be renormalized so that there is a basis $B$ such that
$x_{i}B\subset{\mathbb{Z}}B$ for $i>s$.

Let $d_{i}$ be the filtration degrees of $x_{i}$. We have commutation
relations

$\displaystyle[x_{i},x_{j}]=P_{ij}(x_{1},\dots,x_{r}),$

where $P_{ij}$ is a noncommutative polynomial of degree $\leq d_{i}+d_{j}-2$.
Multiplying $x_{i}$ by $N^{d_{i}}$ for a suitable integer $N\geq 1$, we can
make sure that the coefficients of $P_{ij}$ are in ${\mathbb{Z}}$. Then any
monomial in $x_{i}$ can be written as a ${\mathbb{Z}}$-linear combination of
ordered monomials in which the index is nondecreasing from left to right.

Let $v_{1},\dots,v_{n}$ be a basis of $\tau$. The elements $x_{i}$, $i>s$ act
by some $n$ by $n$ matrices in this basis. By adjusting the integer $N$
introduced above, we may make sure that these matrices have entries in
${\mathbb{Z}}$.

Now let $\tau_{\rm int}=\oplus_{i}{\mathbb{Z}}v_{i}\subset\tau$, and
$\Delta(\tau)_{\rm int}={\mathbb{Z}}\langle x_{1},\dots,x_{s}\rangle\tau_{\rm
int}$. Since homogeneous subspaces of $\Delta(\tau)$ are finite dimensional in
each degree, this group is finitely generated in each degree, hence free, and
$\Delta(\tau)_{\rm int}\otimes_{{\mathbb{Z}}}{\mathbb{Q}}=\Delta(\tau)$.

We claim that $\Delta(\tau)_{\rm int}$ is invariant under $x_{i}$ for any
$i>s$. Indeed, it suffices to show that $x_{i}x_{1}^{m_{1}}\cdots
x_{s}^{m_{s}}v_{j}\in\Delta(\tau)_{\rm int}$. As explained above, we can write
$x_{i}x_{1}^{m_{1}}\cdots x_{s}^{m_{s}}$ as a ${\mathbb{Z}}$-linear
combination of ordered monomials, so it suffices to show that
$x_{1}^{n_{1}}\cdots x_{r}^{n_{r}}v_{j}\in\Delta(\tau)_{\rm int}$. But this
follows from the definition, since $x_{s+1}^{n_{s+1}}\cdots
x_{r}^{n_{r}}v_{j}$ is an integer linear combination of $v_{k}$.

Thus, any homogeneous basis $B$ of $\Delta(\tau)_{\rm int}$ has the required
property $x_{i}B\subset{\mathbb{Z}}B$ for $i>s$. ∎

Thus, Theorem 4.9 for $g$ of finite order is proved. Let us now prove Theorem
4.9 in general. Lemma 4.7 and Corollary 4.8 imply that the orders of the poles
of the rational function ${\rm Ch}_{\tau}({\mathbf{a}},g,t)$ for $g$ of finite
order are uniformly bounded on a Zariski dense set $Z\subset{\mathbf{T}}$ of
such $g$. Thus there exists a finite product $\Pi$ of binomials of the form
$1-\mu$ such that $\Pi(g\nu(t)){\rm Ch}_{\tau}({\mathbf{a}},g,t)$ is a
polynomial in $t$ of a fixed degree for any $g\in Z$. But this product belongs
to ${\mathbb{C}}[{\mathbf{T}}][[t]]$. This implies that $\Pi(g\nu(t)){\rm
Ch}_{\tau}({\mathbf{a}},g,t)\in{\mathbb{C}}[{\mathbf{T}}][t]$. Since this is a
function of $g\nu(t)$, it has the form $P(g\nu(t))$ for some
$P\in{\mathbb{C}}[T]$, as claimed. ∎

######  Remark 4.13.

For ${\mathbf{a}}=1$ Theorem 4.9 follows from the Hilbert syzygies theorem.
Indeed, the commutative algebra $\operatorname{gr}({\mathbf{A}}^{\geq 0})$ is
finitely generated by the argument in [14, proof of Lemma 3.1.2].

Theorem 4.9 implies that it makes sense to set the auxiliary variable $t$ to
$1$ and define the reduced character

$\displaystyle{\rm
Ch}_{\tau}({\mathbf{a}},g):=\operatorname{Tr}_{\Delta(\tau)}({\mathbf{a}}g)=\frac{P(g)}{\prod\limits_{j=1}^{r}(1-\mu_{j}(g))},$

where $P\in{\mathbb{C}}[{\mathbf{T}}]$, which is a rational function on
${\mathbf{T}}$.

Moreover, let ${\mathbf{h}}_{1},\dots,{\mathbf{h}}_{k}$ be a basis of
$\operatorname{Lie}{\mathbf{T}}\subset{\mathbf{A}}$, such that a general
element of ${\mathbf{T}}$ can be written as
$g=\prod\limits_{i=1}^{k}t_{i}^{{\mathbf{h}}_{i}}$. Let $h_{i,\tau}$ be the
eigenvalue of ${\mathbf{h}}_{i}$ on $\tau\subset\Delta(\tau)$. Then

$\displaystyle{\rm
Ch}_{\tau}({\mathbf{a}},g)=\operatorname{Tr}_{\Delta(\tau)}\left({\mathbf{a}}\prod_{i=1}^{k}t_{i}^{{\mathbf{h}}_{i}-h_{i,\tau}}\right).$

This shows that a more natural definition of character is

$\displaystyle\widetilde{\rm
Ch}_{\tau}({\mathbf{a}},g):=\prod_{i=1}^{k}t_{i}^{h_{i,\tau}}\cdot{\rm
Ch}_{\tau}({\mathbf{a}},g)=\operatorname{Tr}_{\Delta(\tau)}\left({\mathbf{a}}\prod_{i=1}^{k}t_{i}^{{\mathbf{h}}_{i}}\right),$

even though this is now in general a multivalued, no longer rational, function
on ${\mathbf{T}}$. We call this function the non-reduced, or ordinary
character of $\Delta(\tau)$ (as it agrees with the usual notions of character
for representations of semisimple Lie algebras and rational Cherednik
algebras). Similarly to Corollary 4.8, one can show that the functions
$\widetilde{\rm Ch}_{\tau}({\mathbf{b}}_{i},g)$ satisfy a holonomic system of
linear differential equations:

$\displaystyle\partial_{l}\widetilde{\rm
Ch}_{\tau}({\mathbf{b}}_{i},g)=\sum_{j=1}^{r}p_{ijl}(g)\widetilde{\rm
Ch}_{\tau}({\mathbf{b}}_{j},g),\qquad l=1,\dots,k,$

where $\partial_{l}$ is the derivative along ${\mathbf{h}}_{l}$ and
$p_{ijl}\in R$ are independent of $\tau$.

Furthermore, such a character can be defined for any object $M\in\mathcal{O}$:

$\displaystyle\widetilde{\rm
Ch}(M)({\mathbf{a}},g)=\operatorname{Tr}_{M}\left(a\prod_{i=1}^{k}t_{i}^{{\mathbf{h}}_{i}}\right).$

If $M=\sum_{\tau}m_{\tau}\Delta(\tau)$ in the Grothendieck group of
$\mathcal{O}$ (a decomposition which always exists and is unique) then
$\widetilde{\rm Ch}(M)=\sum_{\tau}m_{\tau}\widetilde{\rm Ch}_{\tau}$.

####  4.2.4 Examples

######  Example 4.14.

Let $G$ be a finite group, $\mathfrak{h}$ an irreducible finite dimensional
representation of $G$, $V=\mathfrak{h}\oplus\mathfrak{h}^{*}$ and $X=V/G$
(this is a special case of the setting of Section 4.1.1, when
$V=\mathfrak{h}\oplus\mathfrak{h}^{*}$). In this case the symplectic
reflection algebra ${\mathbf{A}}_{c}$ quantizing $\mathcal{O}(X)$ is called
the spherical rational Cherednik algebra. Let $\nu(t)$ act by $t^{-1}$ on
$\mathfrak{h}$ and by $t$ on $\mathfrak{h}^{*}$. Then the only fixed point of
$\nu(t)$ is $0\in X$, so the above analysis applies. Consider a spherical
parameter $c$, i.e., such that
${\mathbf{H}}_{c}{\mathbf{e}}{\mathbf{H}}_{c}={\mathbf{H}}_{c}$ (such
parameters form a nonempty Zariski open set). Then the algebra
$C_{\nu}({\mathbf{A}})$ is Morita equivalent to ${\mathbb{C}}G$, so its
irreducible representations $\tau$ correspond to irreducible representations
$\overline{\tau}$ of ${\mathbb{C}}G$, with
$\Delta(\tau)={\mathbf{e}}M(\overline{\tau})$, where $M(\overline{\tau})$ is
the Verma module over ${\mathbf{H}}_{c}$ with highest weight $\tau$. This
gives a direct proof of Theorem 4.9 (rationality of reduced characters) for
the algebra ${\mathbf{A}}_{c}$ (since a similar statement is easily proved for
${\mathbf{H}}_{c}$ using its triangular decomposition). Also, in this case an
argument similar to the proof of [10, Theorem 1.8] shows that for generic $g$
the dimension of $HH_{0}({\mathbf{A}},{\mathbf{A}}g)$ equals the number of
conjugacy classes in $G$, which implies that the map ${\rm Ch}|_{t=1}$ is an
isomorphism. Thus, all $g$-twisted traces in this case are obtained from
category $\mathcal{O}$. Also, by deforming from $c=0$ we see that these traces
are Weil generically nondegenerate, i.e., give rise to nondegenerate short
star-products. Thus such star-products for fixed $c$ and $g$ depend on $r-1$
parameters, where $r$ is the number of conjugacy classes in $G$.

######  Example 4.15.

Let $X$ be the nilpotent cone of a simple Lie algebra $\mathfrak{g}$, and keep
the notation of Section 4.1.2. Let $\chi$ be a regular central character. Let
$\nu(t)$ be defined by the condition that $\alpha(\nu(t))=t$ for all positive
simple roots $\alpha$. In this case $C_{\nu}({\mathbf{A}})={\rm Fun}(W)$, the
algebra of complex-valued functions on the Weyl group $W$, and the category
$\mathcal{O}$ is the usual BGG category $\mathcal{O}_{\chi}$ for
$\mathfrak{g}$.

Moreover, in this case we can take ${\mathbf{b}}_{i}$ to be polynomials on
$\mathfrak{h}$, so ${\rm Ch}_{\tau}({\mathbf{b}}_{i},g,t)$ are obtained by
differentiating the Hilbert series ${\rm Ch}_{\tau}(1,g,t)$ with respect to
$g$, hence are rational functions of $g\nu(t)$. Thus, by Lemma 4.7, ${\rm
Ch}_{\tau}({\mathbf{a}},g,t)$ is a rational function for all
${\mathbf{a}}\in{\mathbf{A}}$ (giving a direct proof of Theorem 4.9 in this
case). Finally, for generic $g\in{\mathbf{T}}$ the dimension of
$HH_{0}({\mathbf{A}},{\mathbf{A}}g)$ equals $|W|$, as follows, for example,
from [29]. Thus the map ${\rm Ch}|_{t=1}$ in this case is also an isomorphism,
and we again obtain all twisted traces from category $\mathcal{O}$. Finally,
it follows from Section 4.1.2 that Weil generically these traces are
nondegenerate, hence define nondegenerate short star-products. So we see that
nondegenerate short star-products (for fixed $\lambda$ and $g$) depend on
$|W|-1$ parameters. Thus, the total number of parameters for such star-
products (up to conjugation by the group $G$) is
$2\operatorname{rank}(\mathfrak{g})+|W|-1$.

A similar analysis applies to the case when $X$ is a Richardson orbit in
$\mathfrak{g}^{*}$, as in Remark 4.5. In this case, instead of Verma modules
we should use parabolic Verma modules.

######  Remark 4.16.

Note that in Examples 4.14, 4.15 we have $s\in{\mathbf{T}}$, hence $s$ acts
trivially on $HH_{0}({\mathbf{A}},{\mathbf{A}}g)$ and the $s$-invariance
condition for twisted traces is vacuous.

######  Example 4.17.

Let $X$ admit a symplectic resolution $\widetilde{X}$, and ${\mathbf{T}}$ act
on $\widetilde{X}$ with $N<\infty$ fixed points, hence $X^{{\mathbf{T}}}=0$
(the situation of [3]). For simplicity assume that $s\in{\mathbf{T}}$. This
includes Example 4.15 ($\widetilde{X}=T^{*}G/B$, the Springer resolution), and
also Example 4.14 for $G=S_{n}\ltimes\Gamma^{n}$ where $\Gamma\subset{\rm
SL}_{2}({\mathbb{C}})$ is a finite subgroup. In this case, it follows from [3]
that for generic $\lambda$ the algebra $C_{\nu}({\mathbf{A}})$ is a
commutative semisimple algebra of dimension $N$, and for generic $g$ we have
$\dim HH_{0}({\mathbf{A}},{\mathbf{A}}g)=N$, so the map ${\rm Ch}|_{t=1}$ is
an isomorphism. Thus, traces in representations from category $\mathcal{O}$
give rise to all the $g$-twisted traces.

Also by localizing to the fixed points it can be shown in this case that the
Weil generic $g$-twisted trace is nondegenerate. Thus, in this case for
generic $\lambda$, $g$ we obtain a family of nondegenerate short star-products
with $N-1$ parameters. This will be discussed in more detail in a subsequent
paper.

### Acknowledgements

The first author is grateful to C. Beem and A. Kapustin for introducing him to
the problem, M. Kontsevich for contributing a key idea, and C. Beem, M.
Dedushenko, D. Gaiotto, D. Kaledin, D. Kazhdan, I. Losev, G. Lusztig, H.
Nakajima, E. Rains, L. Rastelli and T. Schedler for useful discussions. The
work of the first author was partially supported by the NSF grant DMS-1502244.
The work of the second author was supported by the MIT UROP (Undergraduate
Research Opportunities Program).

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