Various objects in Lie theory have the combinatorics of their representations described in terms of roots and weights; the structure of Weyl chamber walls is crucial for many criteria in their study. Certain important functors in this representation theory cross the chamber walls. We talk about the wall crossing functors in representation theory.

Symmetries related to Weyl groups, Weyl chambers and chamber walls are involved (what is sometimes also in the other notion of wall crossing in the BPS setup). A priori wall crossing functors in representation theory (introduced in 1970s by the Moscow school, Gelfand, Bernstein etc.) are about certain functors which in take as input an infinite-dimensional representation, tensor it with finite-dimensional and look for certain pieces in their decomposition. In this study, the position with respect to the chamber walls is crucial.

- A. Beilinson, V. Ginzburg,
*Wall-crossing functors and $D$-modules*, Representation Theory**3**(electronic), 1–31 (1999) pdf.

The composition of the translation functor that sends the category at a regular maximal ideal to the category at a non-regular maximal ideal with the translation functor acting in the opposite direction is called a wall-crossing functor.

- J. Bernstein, S. Gelfand,
*Tensor products of finite and infinite dimensional representations of semisimple Lie algebras*, Comp. Math. 41 (1980), 245–285. MR 82c:17003

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