There was not standard issue weapon that was issued to cavalry units in general and for a specific battalion or regiment you would need to look at the quarterly ordnance returns or if one of the soldiers note what they were issued at the time of formation that would be worthy of note.
Arming all regiments in Georgia was met with varying degrees of difficulty depending on what time period of the war it was. You may want to initially check the Compiled Military Service Records (CMSR) of the officers (I would start with the field grade officers) in the regiment and see if they had receipts for issuance of weapons or accouterments that may give you an idea of what they initially had. The next step would be to check the CMSR's of the Brigade Staff officers to see if they have quarter ordnance reports or receipts in their CMSR's.
Please let me know if you have any questions.
Gerald D. Hodge, Jr.
War Between the States Historian
Historian: 39th Georgia Volunteer Infantry Regiment