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Okay, thank you.

So yesterday I explained to you a little bit about the localization formulas of Disharmut and Heckman for ordinary moment maps.

So today I want to give you something, I want to explain something about the localization for group-valued moment maps.

Now, let me start by the following introduction.

Remember, in comparison to ordinary moment maps, to look at group-valued moment maps we need one additional structure.

And this additional structure, it's a scalar product on the Lie algebra of our group.

So in other words, we have this Lie algebra G, and so it should be quadratic.

Let me recall what this means.

So we need an invariant scalar product on the Lie algebra.

So invariant scalar product.

And well, invariance here is the bracket of x, y scalar product with z plus y bracket of x, z is 0 for all x, y and z in G.

So in fact, somewhat surprisingly, this construction, it gives a lot of extra structure.

So once you have a Lie bracket plus the scalar product, you can do a lot of things.

Now I will give you some short list of things which will be useful in the talk.

So on my list, which is far from being complete, there are about seven items.

So let me show you what happens when you have a Lie structure together with invariant scalar product.

So one thing which is quite obvious, now the dual space to your Lie algebra and the Lie algebra itself are naturally identified.

And moreover, they are identified as G representations.

That's because of this invariance.

So the adjoint action and the quadjoint action are now the same.

So this space is identified and the adjoint and quadjoint actions are the same.

Now, well, what was my second point?

So the second point you already seen in the talks of Pavel Severo.

So there is a canonical element in the third exterior power of G star.

So it's given by the formula eta of x, y, z equal to, say for instance, a bracket of x, y scalar product with z.

So that's more or less the only combination that we can come up with in this context.

So the same combination as enters this invariance condition.

So this is called the Kortan 3-form.

In fact, with this Kortan 3-form you already seen one can also give it another interpretation.

So one can think of eta as an element of omega 3 of G.

So as a 3-form on G.

So this 3-form is by invariance.

So invariant on the left and on the right.

And I think Pavel showed that this implies the closedness.

So d eta is equal to 0.

Well, now some more stuff which comes from this scalar product.

So G is a vector space and once you have a scalar product one can define the Clifford algebra based on this vector space.

So the Clifford algebra.

And let me just recall this means an algebra with those defining relations.

So x, y plus y, x equals twice this scalar product.

In fact, sometimes it is convenient to scale it.

So if you are into quantum theory or into deformation quantization you can put here deformation parameters say h bar.

So sometimes it comes useful.

I'll probably think that h bar is equal to 1 for the most of my talk.

But sometimes it is convenient to have it.

Now, well, one remark about this Clifford algebra.

You know that the Clifford algebra as a vector space that's the same thing as the exterior algebra.

But the product is different.

But product is not the same.