In mathematics, the affine Grassmannian of an

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...

''G'' over a field ''k'' is an

ind-scheme In algebraic geometry, an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes. Examples *\mathbbP^ = \varinjlim \mathbbP^N is an ind-scheme. *Perhaps th ...

—a colimit of finite-dimensional schemes—which can be thought of as a

flag variety In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smo ...

for the loop group ''G''(''k''((''t''))) and which describes the representation theory of the

Langlands dual In representation theory, a branch of mathematics, the Langlands dual ''L'G'' of a reductive algebraic group ''G'' (also called the ''L''-group of ''G'') is a group that controls the representation theory of ''G''. If ''G'' is defined over a fie ...

group ^''L''''G'' through what is known as the geometric Satake correspondence.

Definition of Gr via functor of points

Let ''k'' be a field, and denote by

k\text

and

\mathrm

the category of commutative ''k''-algebras and the category of sets respectively. Through the

Yoneda lemma In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (vie ...

, a scheme ''X'' over a field ''k'' is determined by its

functor of points In algebraic geometry, a functor represented by a scheme ''X'' is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme ''S'' is (up to natural bijections) the set of all morphisms S \to X. T ...

, which is the functor

X:k\text \to \mathrm

which takes ''A'' to the set ''X''(''A'') of ''A''-points of ''X''. We then say that this functor is representable by the scheme ''X''. The affine Grassmannian is a functor from ''k''-algebras to sets which is not itself representable, but which has a filtration by representable functors. As such, although it is not a scheme, it may be thought of as a union of schemes, and this is enough to profitably apply geometric methods to study it. Let ''G'' be an algebraic group over ''k''. The affine Grassmannian Gr_''G'' is the functor that associates to a ''k''-algebra ''A'' the set of isomorphism classes of pairs (''E'', ''φ''), where ''E'' is a

principal homogeneous space In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...

for ''G'' over Spec ''A'' and ''φ'' is an isomorphism, defined over Spec ''A''((''t'')), of ''E'' with the trivial ''G''-bundle ''G'' × Spec ''A''((''t'')). By the

Beauville–Laszlo theorem In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve. It was proved by . The theorem ...

, it is also possible to specify this data by fixing an

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...

''X'' over ''k'', a ''k''-point ''x'' on ''X'', and taking ''E'' to be a ''G''-bundle on ''X''_''A'' and ''φ'' a trivialization on (''X'' − ''x'')_''A''. When ''G'' is a

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...

, Gr_''G'' is in fact ind-projective, i.e., an inductive limit of projective schemes.

Definition as a coset space

Let us denote by

\mathcal K = k((t))

the field of

formal Laurent series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...

over ''k'', and by

\mathcal O = k t

the ring of formal power series over ''k''. By choosing a trivialization of ''E'' over all of

\mathrm\mathcal O

, the set of ''k''-points of Gr_''G'' is identified with the coset space

G(\mathcal K) / G(\mathcal O)

References

*{{cite book, author=Alexander Schmitt, title=Affine Flag Manifolds and Principal Bundles, url=https://books.google.com/books?id=xrPoBAdiVdQC&pg=PA1, accessdate=1 November 2012, date=11 August 2010, publisher=Springer, isbn=978-3-0346-0287-7, pages=3–6 Algebraic geometry