The notion of a Kazhdan-Lusztig theory for a general highest weight category was introduced by the authors [in Tôhoku J. Math., II. Ser. 45, No. 4, 511-534 (1994; Zbl 0801.20013); Proc. Symp. Pure Math. 56, Pt. 1, 63-104 (1994; see the preceding review Zbl 0819.20044); Contemp. Math. 139, 43-73 (1992; Zbl 0795.20022)] and they proved that the existence of such a theory is sufficient to calculate the \(\text{Ext}^ \bullet\)- groups between simple objects in terms of “Kazhdan-Lusztig” polynomials.
In the present paper the authors study the algebra structure induced by Yoneda multiplication on these \(\text{Ext}^ \bullet\)-groups. Let \(\mathcal C\) be a highest weight category with finite weight poset such that the objects in \(\mathcal C\) have finite length. Let \(L\) denote the direct sum of the distinct irreducible objects in \(\mathcal C\), and \(A^ ! = \text{Ext}^ \bullet_{\mathcal C}(L,L)\). The module category \({\mathcal C}^ !=\text{mod- }A^ !\) is referred to as the homological dual of \(\mathcal C\).
The first main result of this paper establishes that, if \(\mathcal C\) has a Kazhdan-Lusztig theory, then \({\mathcal C}^ !\) is also a highest weight category. Further properties of \({\mathcal C}^ !\) are studied, assuming that \(\mathcal C\) has a Kazhdan-Lusztig theory. It is shown that the stratifications defined by chains of ideals in the weight poset of a highest weight category, dualize perfectly, with strata dual to those in the original stratification. Also the question is taken up of when the homological dual itself possesses a Kazhdan-Lusztig theory: Theorem 3.8 gives a simple sufficient condition for this to occur, and Theorem 3.9 provides a partial converse.