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Another motivation is more basic. For example, in the modular representation theory of finite groups it is often the case that one has two blocks $A$ and $B$ of two different groups and one suspects (for example) that both blocks have the same number of simple modules (an example of this is given by Alperin's conjecture).

Now, suppose that $A$ and $B$ are Morita equivalent. Then not only do $A$ and $B$ have the same number of simple modules, but specifying a Morita equivalence gives a bijection between the simple modules.

Often, however, $A$ and $B$ are not Morita equivalent, but rather derived equivalent. The Grothendieck groups of $D^b(A)$ and $D^b(C)$ have a basis given by the classes of the simple modules of $A$ and $B$. A derived equivalence induces an isomorphism between the Grothendieck groups, and hence $A$ and $B$ have the same number of simple modules if they are derived equivalent. Note now, however, that the derived equivalence does not induce a bijection between simple modules, because simple modules need not correspond under the derived equivalence.

As an example of this approach is Broué's abelian defect group conjecture (which predicts a derived equivalence between certain $A$ and $B$). It implies Alperin's conjecture (in the abelian defect case), but provides a structural reason for the equality (and also implies much more structural results about characters, "perfect isometries" ...).

Hence, one may search for a derived equivalence to give a structure explanation for various concrete numerical equalities.

(I think another example of this is given by Bezrukavnikov's use of perverse coherent sheaves on the nilpotent cone to explain some numerical equivalences observed by Vogan, but I don't know much about this.)

2 added 15 characters in body

Another motivation is more basic. For example, in the modular representation theory of finite groups it is often the case that one has two blocks $A$ and $B$ of two different groups and one suspects (for example) that both blocks have the same number of simple modules (an example of this is given by Alperin's conjecture).

Now, suppose that $A$ and $B$ are Morita equivalent. Then not only do $A$ and $B$ have the same number of simple modules, but specifying a Morita equivalence gives a bijection between the simple modules.

Often, however, $A$ and $B$ are not Morita equivalent, but rather derived equivalent. The Grothendieck groups of $D^b(A)$ and $D^b(C)$ have a basis given by the classes of the simple modules of $A$ and $B$. A derived equivalence induces an isomorphism between the Grothendieck groups, and hence $A$ and $B$ have the same number of simple modules if they are derived equivalent. Note now, however, that the derived equivalence does not induce a bijection between simple modules, because simple modules need not correspond under the derived equivalence.

As an example of this approach is Broué's abelian defect group conjecture (which predicts a derived equivalence between certain $A$ and $B$). It implies Alperin's conjecture, but provides a structural reason for the equality (and also implies much more structural results about characters, "perfect isometries" ...).

Hence, one may search for a derived equivalence to give a structure explanation for various concrete numerical equalities.

(I think another example of this is given by Bezrukavnikov's use of perverse coherent sheaves on the nilpotent cone to explain some numerical equivalences observed by Vogan, but I don't know much about this.)

1

Another motivation is more basic. For example, in the modular representation theory of finite groups it is often the case that one has two blocks $A$ and $B$ of two different groups and one suspects (for example) that both blocks have the same number of simple modules (an example of this is given by Alperin's conjecture).

Now, suppose that $A$ and $B$ are Morita equivalent. Then not only do $A$ and $B$ have the same number of simple modules, but specifying a Morita equivalence gives a bijection between the simple modules.

Often, however, $A$ and $B$ are not Morita equivalent, but rather derived equivalent. The Grothendieck groups of $D^b(A)$ and $D^b(C)$ have a basis given by the simple modules of $A$ and $B$. A derived equivalence induces an isomorphism between the Grothendieck groups, and hence $A$ and $B$ have the same number of simple modules if they are derived equivalent. Note now, however, that the derived equivalence does not induce a bijection between simple modules, because simple modules need not correspond under the derived equivalence.

As an example of this approach is Broué's abelian defect group conjecture (which predicts a derived equivalence between certain $A$ and $B$). It implies Alperin's conjecture, but provides a structural reason for the equality (and also implies much more structural results about characters, "perfect isometries" ...).

Hence, one may search for a derived equivalence to give a structure explanation for various concrete numerical equalities.

(I think another example of this is given by Bezrukavnikov's use of perverse coherent sheaves on the nilpotent cone to explain some numerical equivalences observed by Vogan, but I don't know much about this.)